P De Buhan1

A Corfdi�

LIMIT DESIGN OF AXISYMMETRIC SHELLS WITH APPLICATION TO CELLULAR COFFERDAMS

This pape � is devoted to the limit design of cellular cofferdams that are regarded as mixed struc tures where th� ba � kfill 1s model�d as � thre_e-dimensional continuum, while the surrounding sheet pile wall is treated as a . cylin _ drical s�ell. Deahng w1� this structure from a static point of view, it turns out that the problem under . consideration requ1res the calculation of the ultimate load value of a circular cylindrical shell subjected to a lm � arly varying pressure distrib � tion rep � esenting th � thrust of the backfill material. Extending the results of prev10u � works, a com � lete solution to this .

problem IS developed for different boundary conditions. The cor : espondm � results are discussed, notably the � � �uence of the shell relative thickness. They are applied to the des1gn of a smgle cell � lar c c;> fferdam _ whose stability under gravity forces is examined, with the strength of the granular backfill matenal bemg descnbed by a Mohr-Coulomb criterion.

INTRODUCTION

Cellular cofferdams are civil engineering structures that are most commonly encountered in hydraulic works, where they provide a convenient means for isolating dry work sites in water-covered areas. They may also be used as breakwaters or retaining walls. They are constructed by assembling vertical metallic sheet piles, either driven into the soil or simply resting on it, in order to form circular cylindrical cells filled with a granular material. Although the intensive use of these kinds of structures is not new (it may even be traced back to the beginning of the century), their mechanical behavior is still to be fully understood, particularly their limit design.

Simplified engineering design rules are already available (see, for instance, [START_REF] Bowles | Foundation analysis and design[END_REF] or [START_REF] Lacroix | Design, construction, and performance of cellular cofferdams[END_REF], based on the idea that the thrust of the backfill material inside the cell generates interlocking tensions between the sheet piles, to which the latter should be able to resist. One of the difficult points is to decide which value of the backfill thrust should � taken into account in such an analysis. A first comprehen Sive approach that provides a rigorous mechanical framework to this problem has recently been developed by Dormieux and Delaurens (1991), whose analysis is based on the yield design (or limit design) theory [START_REF] Salen�on | Calcul a la rupture et analyse limite[END_REF][START_REF] Salen�on | An introduction to the yield design theory and its application to soil mechanics[END_REF][START_REF] Salen�on | Yield design: a survey of the theory[END_REF]. Making use of both the lower bound and upper bound methods, they derived crude but rigorous estimates for the stability factor of a single cell subjected to its own weight. In regard to the lower ?ound estimates that they obtained through the static approach, 1t should be noted that their calculation implicitly amounts to disregarding any resistance of the sheet piles to bending mo ments. It thus appears that a possible way to improve these estimates consists of taking the bending resistance into ac count. It is, therefore, necessary to get a better insight into the mechanical behavior of the sheet pile cell, which from now on will be modeled as a homogeneous cylindrical shell.

Limit analysis methods applied to plates and shells, re garded as two-dimensional continuous media, have formed the subject of many prominent works, among which one should quote those of [START_REF] Drucker | Limit analysis of cylindrical shells under axially symmetric loading[END_REF], [START_REF] Hodge | The rigid-plastic analysis of symmetrically loaded shells[END_REF][START_REF] Hodge | Limit analysis of rotationnally symmetric plates and shells[END_REF], [START_REF] Onat | Limit analysis of shells of revolu tion[END_REF], [START_REF] Eason | The influence of free ends on the load carrying capacities of cylindrical shells[END_REF], and [START_REF] Drucker | Limit analysis of symmetri cally loaded thin shells of revolution[END_REF]. [START_REF] Drucker | Limit analysis of symmetri cally loaded thin shells of revolution[END_REF], for instance, pro posed a simplified yield strength criterion for shells, involving no interaction between normal forces and bending moments. This criterion may be regarded as an approximation to the true strength criterion for a shell made of a homogeneous Tresca material. Using such a criterion, [START_REF] Olszak | Die grenztragfahigkeit von zylin drischen schalen bei verscheidenen formen der plastizitatsbedingung[END_REF] have derived solutions to the classical problem of a tank sub mitted to a hydrostatic distribution of pressure. These results have been reported, with slight modifications, in Save and M _ assonnet (1972). Unfortunately, they cannot be directly ap phed to cellular cofferdams, since the analysis performed by [START_REF] Olszak | Die grenztragfahigkeit von zylin drischen schalen bei verscheidenen formen der plastizitatsbedingung[END_REF] in the relevant situation (tank with free upper edge) was restricted to "short" shells; that is, to shells whose length is small compared to their radius (given a value of the thickness).

The main purpose of the present paper is to derive exact solutions to the aforementioned problem, for short as well as long cylindrical shells. Three kinds of boundary conditions conc � ming the lower edge of the shell will be successively exammed. They are representative of different situations one may come across in practice, depending on the construction conditions of the cellular cofferdams. These solutions are first obtained by means of the yield kinematic approach, making use of failure mechanisms involving hinge circles, then by exhibiting generalized stress distributions along the shell, in order to prove that the lower and upper bound estimates co i � cide. Th � results so � btained make it possible to quite sig mficantly Improve prev10us lower bound estimates for the sta bility factor of cellular cofferdams.

STABILITY ANALYS IS OF CELLUL AR COFFERDAM: STATEMENT OF PROBLEM

Cellular cofferdams are civil engineering structures com prised of a cylindrical shell made of metallic sheet piles con nected to each other along the vertical direction filled with a � ohesionl � ss granular material (sand or gravel) [Fig. 1 (a)]. It 1s co � vement to analyze the stability of such a structure by resortmg to _ a so-called "mixed modeling" approach where the surroundmg sheet pile wall is schematized as a cylindrical shell, while the backfi ll material is considered as a three-di mensional continuous medium whose strength properties are defined through a Mohr-Coulomb condition that writes f(a) = (a, -<1 3 ) + (a, + a3)sin 'P s 0

(1)

where a = stress tensor at any point of backfill material; a1 �d <13 =major and minor principal stresses, respectively (ten sile stresses are counted as being positive); and 'P = friction angle. It is thus assumed that the backfill material exhibits no cohesion.

The whole structure is resting on a horizontal rigid bedrock.

It is referred to as a cylindrical coordinate system (r, 6, z) with the origin being taken as the intersection of the lower plane with the symmetry axis. No surcharge is applied to the struc ture apart from the gravity characterized by a uniform field of body forces equal to -"Ye., where "Y is the specific weight of the backfill material [Fig. l(b)]. Concerning the sheet pile wall, modeled as a cylindrical shell of height H and radius R, it will be assumed that no external load is applied on its upper edge (z = H), while the following three kinds of boundary conditions will be considered in the sequel for the lower edge (z = 0):

• Free edge-the lower end is free to move horizontally as well as to rotate

• Simply supported edge-the displacements are fixed equal to zero, while the rotation remains free

• Clamped edge-no translation and rotation are allowed These boundary conditions may be considered as modeling, in a realistic way, the physical interaction between the sheet piles and the substratum. By way of illustration, the third boundary condition (clamped edge) corresponds, for instance, to the case when the sheet piles are driven deep enough into the substratum, while the free edge condition would be asso ciated with negligible horizontal friction forces between the sheet pile extremities and the substratum. The second type of boundary condition represents an intermediate situation.

Ta king into account the axisymmetry of the problem, it is advisable to consider distributions of stresses in the shell that also satisfy this symmetry.

In terms of normal and shear resultants as well as bending moments (referring to an infinitesimal shell element such as that shown in Fig. 2), the following components vanish: ( 2 a-c)

Resorting to the same argument of axisymmetry, it appears that the stress vector T applied at any point of the shell by the backfill material is of the form

T = p(z)e, + T( z)e.
(3)

Enforcing both the resultant and moment equilibrium of the shell element shown in Fig. 2 leads to the following equations:

dV . N.. dM ZZ dz -R + p(z) = 0; dz + v. = 0 dN .. dM 00 dz + T(Z) = 0; de = 0 (4a, b) (4c, d)
Eliminating v. between (4a) and (4b), the following clas sical second-order differential equation is obtained:

d2 M zz Noo dz 2 + R -p(z) = 0 (5)
which may be found, for instance, in [START_REF] Chen | Plasticity for structural engineers[END_REF]. According to the yield design theory (or limit design) for mulation [START_REF] Salen�on | Calcul a la rupture et analyse limite[END_REF][START_REF] Salen�on | An introduction to the yield design theory and its application to soil mechanics[END_REF][START_REF] Salen�on | Yield design: a survey of the theory[END_REF], the stability of the whole structure is ensured if one can exhibit a stress distri bution of membrane forces (N ", N99) and moments (M66, M") in the shell, with both of them complying with equilibrium and strength requirements. Dealing with this problem by means of the static approach, a particular choice is now being made concerning the stress fields in the backfill material, de fined as follows in the cylindrical coordinates:

CJ'zz = -y ( z -H), CJ'rr = CJ'eo = li.CJ".. and

(7a)

CJ"11 = 0 for the other components where Ka = (1 -sin <p)/(1 + sin <p) and K P = K:; 1 denote classical active and passive earth pressure coefficients, respec tively.

The corresponding stress distribution exerted on the sur rounding shell is therefore It then remains to be seen whether the shell will be able to resist the linearly varying pressure distribution (10). This will now be considered in itself as an auxiliary yield design prob lem, where P stands for loading parameter.

T = a•(-e,) = -X:y(z-H)e,

YIELD DESIGN FORMULATION OF AUXILIARY PROBLEM

The auxiliary problem under consideration is that of a cy lindrical shell of height H and radius R subjected to an internal pressure distribution increasing linearly from zero at the top of the shell to a prescribed value denoted by P at the bottom (Fig. 3). As a result of (6) with z = 0, it appears that N,, = 0, so that the only stress components having nonzero values are Nee and M,, which fulfill equilibrium (5), while from (4d) Mee is an arbitrary function of z. Following Olszak and Saw czuk's (1959) approach, a simplified criterion limiting the val ues of Nee and M,,, respectively, will be adopted

(l1a,b)
Such a criterion may be considered as an approximation of the actual strength criterion, which can be derived in the case of a shell made of a homogeneous Tresca material [START_REF] Drucker | Limit analysis of symmetri cally loaded thin shells of revolution[END_REF]. More specifically, denoting by <Yo the resis tance of the constituent material under a uniaxial stress, it can be shown that (11) is an upper bound of the exact criterion if one takes N0 = cr0e and M0 = cr0e2/4 (e is the thickness of the shell), while it turns out to be a lower-bound estimate when N0 = (3/4)cr0e and M0 = (3/16)croe2• In the case of a cellular cofferdam, the shell being constituted by an assemblage of metallic sheet piles, it seems reasonable, in the absence of more specific data, to adopt a simple criterion as (11), taking for N0 the force per unit length along the connection between two sheet piles necessary for tearing them apart. This force can be measured through a very simple testing procedure.

Within the framework of the yield design theory, it appears that the ultimate load p+ is defined as the maximum value of P for which one can exhibit at least one distribution of Nee and M .. As it is quite apparent from the previous definition.!+ may be written as

(19)
that is, as a function of the sole parameter k.

Lower Bound Static Approach

This approach simply consists of applying the definition of f+; that is, in exhibiting particular classes of stress distribu tions that satisfy both equilibrium and strength requirements. As a first very simple approach, one may consider such dis tributions where the bending moment is taken to be equal to zero [m(u) = 0]. Eq. (13) then immediately gives

n(u) = f(l -u) (20)
and the strength condition (14), which reduces here to ln(u)l s 1 u E [0, 1 ], implies that f ::5 1.

It follows that

(21)
for any value of k � 0.

Upper Bound Kinematic Approach

This approach results from the dualization of the static ap proach by means of the virtual work principle, which makes use of kinematically admissible velocity fields for the problem. Any such field that respects the condition of axisymmetry of the problem is characterized in each point of the shell by a set of two kinematic variables. These variables are the component of the velocity along the radial direction, denoted by v, and the rate of rotation w of the normal to the shell at the same point, counted positive along the orthoradial direction. Owing to the symmetry of the problem, v and w are functions of the z-coordinate only (Fig. 4). Note that these two variables are a priori independent.

The principle of virtual work states that, given any statically admissible stress distribution of Nee. M, .. and V, in the shell, the following equality holds:

W,(v, w) + W,(v, w) = 0 whatever (v, w) (22)
W;(v, w) = 2'TT R LH [ Vz (�;-w ) + Nos � + Mzz �:] d z Introducing { ( d v ) v dw ll( v , w) = sup Vz d z -w + N •• R + Mzz d z ;
( d v ) l v l l"l v , w = d z = No R + Mo v for w = d v = v ' (Kirchhoff-Love condition) d z (27) (28)
ll(v, w) = +oo otherwise; i.e., when w -:1-v' since the adopted strength condition involves no limitation on the shear com ponent v •.

The calculation of wmro and hence the kinematic approach, can be extended to velocity fi elds exhibiting discontinuities of both v and w, denoted by [ v ] 1 and [w) 1 , respectively, along a finite number of points along the vertical direction (i = 1-n).

It can be shown that the following quantity is to be added to the integral (27):

.

L ll([ v ] 1 , [w] 1 )
[ v ] 1 , [w] 1 ) = + oo if [ v ] 1 -:1-0 (30) (31a) ll([ v ] 1 , [w] 1 ) = ll([ v ') 1 ) = M0 l[ v 'Jd if [ v ]; = 0 (31b)
taking into account the fact that the Kirchhoff-Love condition holds on both sides of the discontinuity. This means that the only relevant discontinuities are such that v remains continu ous, whereas its fi rst derivative (which coincides with the rate of rotation) may undergo jumps ("hinges") across horizontal circles drawn on the shell surface. Thus

Wm,(v) = 2'TT R {LH ( � o I v i + Mo l v "l ) d z + t Mo l[ v ']; j } (32)
Since W,( v ) = 21TRP f� (1 -z/H)v dz, one finally gets from (26) with P = p+

(33)
provided that f� (1z/H)v dz be strictly positive. The in equality in ( 33) clearly shows that the kinematic approach pro vides upper-bound estimates for the ultimate load p+.

CASE OF CYLINDER WITH FREE LOWER EDGE

This case will fi rst be dealt with by means of the kinematic approach, then through the static approach, thus obtaining the exact solution for the problem.

Kinematic Approach

The class of velocity yields v which we will consider is shown in Fig. 5. [O 11

f s = f , (u�> k) w1th u1 = z1/H u1 E , 1 -u1/3 (36)
The minimal value of the right member, that is, the optimal upper-bound value off+ derived from this particular class of mechanisms, is obtained for u, = ur such that hence

* ' k = u , 6(2 -ur) (37) 2 r :S n[uf(k), k] = /�(k) with f�(k) = 2 -ur (38)
Consequently, the optimal upper bound is given in the fol lowing implicit form as the solution of:

2 (f� -1) 3 k = 3 (/�) 2 (39)
This approach is completed by considering another class of mechanisms where no circular hinge is involved, as shown in Fig. 6.

The motion is defined by

v(z) = a ( 1 -t ) for 0 s z s H (40)
Performing the same calculations as before, one obtains

Wm,(v) = 27raN0H [ u1 -( 1 -2 �J J if U1 :S 1 (41)
We get the following upper bound: ing with the strength criterion everywhere. This can be achieved by using the theorem of association [START_REF] Salen�on | Calcul a la rupture et analyse limite[END_REF] as a guideline. More specifically, in the present case, this the orem states that if any such stress distribution exists, it nec essarily satisfies Nee = + N0 (or n = + 1) in the lower part of the structure, since the associated strain rate component viR in the optimal mechanism is positive. Therefore, the equilib rium (13) combined with the boundary conditions (16) gives

!+ s 3 (2u f -2 u 1 + 1) = /"( k) (42) (3u1 -1) 2 u "
1 [ u2 u 3 ] m(u < ur) = m-(u) = k (f" -1) 2 -f" 6 (45)
It can be easily seen that this distribution is compatible with the strength condition <lm-1 s 1). It may be completed in the upper part (u :2: un by two kinds of distributions.

Sinusoidal Distribution

We consider the distribution defined by ,. 2 2 cos (,. u-ur )

m+(u) =-cos 7r ---1 + 1 1 [ ( u-u * ) J
2 (1 -ur) 1 -ur (47)
A sufficient condition for this distribution to obey the strength condition <In + 1 s 1, ur s u s 1) is

2 (/" -1)(2 -/") 2 k < --"'-----'--'--: :----''---'- -,. 2 (/") 2 (48)
that is, taking into account the relationship between f" and k k :S 0.01629 (4 9)

It follows that F(k) =f"(k) for ks 0.01629 (50)

Polynomial Distribution

This kind of distribution is obtained by choosing n + = a = ct, so that integrating the equilibrium (13) and using the boundary conditions for u = 1 The same kind of approach as that previously developed for a free bottom edge is followed. The failure mechanisms ex plored in the kinematic approach are represented in Fig. 9. They involve up to three hinge circles, with the lowest one, located at the bottom of the cylinder, implying no contribution to the maximum resisting work for a simply supported edge. They produce upper-bound estimates for f+( k), which can be proved to be the exact solution by exhibiting associated stress distributions given in Appendices I and II. As an example, such a distribution, along with the associated failure mecha nism, is shown in Fig. 10 for the case of a simply supported cylinder.

+ 1 [ n 3 a 2 ] m =--(1 -u) --(1 -u) k 6 2 (51)
The results of the present study are gathered in Fig. 11, in the form of curves giving the exact value off+ as a function of k for the three different boundary conditions.

As could be anticipated, the greatest value of f+( k) is ob tained for the case of a clamped edge, the intermediate curve corresponding to the simply supported edge which remains free to rotate about the orthoradial axis.

BACK TO INITIAL PROBLEM

Resuming the analysis of the cellular cofferdam undertaken in an earlier section of the present paper, it appears that the stability of the composite structure (backfill material + sheet pile wall) under its own weight is ensured provided that 'YHR I N0 ::5 ("/HR ! No t

CONCL UDING REMARKS

By adopting a simplified expression for the yield criterion of an axisymmetrically loaded shell expressed in terms of nor mal forces and bending moments, exact solutions to the prob lem of a cylindrical shell submitted to a linearly varying in ternal pressure have been developed. In particular, it has been shown that the ultimate load increases with a parameter k, which characterizes the relative thickness of the shell. More over, the importance of the boundary conditions prescribed at the lower edge of the shell has been clearly pointed out. As an example, the improvement of the lower-bound estimate for the stability factor of a single cellular cofferdam resulting from such an analysis may range between 5% and 12% when com pared to previous conservative estimates where the resistance to bending moments is neglected. It should be emphasized that such improved lower-bound estimates remain valid for any strength condition governing the interface between the sheet pile wall and the backfill material. In other terms, taking into account nonzero shear stresses T along this interface and, hence, more complex stress fields in the backfill zone than those considered in the prior analysis (namely with rr,8 -:;:. 0) would necessarily result in deriving still better lower bounds. Likewise, upper-bound estimates for the stability factor could be expected from the kinematic approach. This would neces sitate constructing relevant velocity fields both in the backfill material and in the surrounding shell. {< 6-36f + 27/2 + 3!'-6v'6Vi+3t2 + 6v'6Vi+3 n} . +�[4(/ + 3)312 -4v6(f + 3) -l2v;+3 + 13v'6]312 (67)

  FIG. 1. Mixed Modeling of Cellular Cofferdam: (a) Basic Struc ture; (b) Coordinate System

  FIG. 2. Equilibrium of Infinitesimal Shell Element under Axi symmetric Conditions

  Integrating (4c) over the height of the shell and taking into account the boundary conditions on the upper edge [N "(z = H) = 0] yields Nzz = -[ T(z ' ) dz' (6) Eqs. (5) and (6) express the equilibrium of the shell subject to the stress distribution p(z) and T(z).

  where ll. represents an arbitrary constant.These stress fields automatically satisfy the equilibrium equation (diva -"Ye, = 0) as well as the boundary conditions prescribed at z = H (stress-free top level). Moreover, it com plies with the strength condition (1) if and only if (8)

  with P = 'A:yH representing maximum pressure applied to bot tom of shell (z = 0).

  FIG. 3. Auxiliary Problem-Cylindrical Shell Submitted to Linear Distribution of Internal Pressure

FIG. 4 .

 4 FIG. 4. VIrtual Velocity Fields Considered for Problem where (v, w) denotes any kinematically admissible velocity field; and w. = work done by external forces W. ( v , w) = 2'TT R LH p( z ) v ( z) d z

  with v .. N88, and Mzz satisfying (11that a necessary condition for the structure to re main stable (i.e., P :S P+) is W.(v, w) ::5 Wm,( v , w) whatever ( v , w) (26) where Wmr(v, w) represents maximum resisting work defi ned as with n

  where ll([ v ] 1 , [w] ;) = sup{Vz[V]; + Mzz[W]; ; with v .. Mw and N88 verifying ( 11)}

  FIG. 5. Failure Mechanism with Circular Hinge

Fig. 7 FIG. 7 .

 77 Fig. 7 represents the giving the variations of both upper bounds f� and /2 as functions of parameter k. It follows that the best upper-bound estimate that can be derived from the two previous kinds of mechanisms is r :S f"(k) = min{f�./2} withf"(k) = /�(k) solution of (39) if k s 0.0276 and/2 ""' 1.4415 otherwise Static Approach

  This distribution verifies I m + I < 1, as well as the continuity of m and m' at the connecting point ur with mand the bound ary conditions on the upper edge [m(l) = m'(1) = 0]. Then, the equilibrium (13) makes it possible to calculate the corre sponding distribution of n, denoted by n + n+(u) =f"( l -u) + !5.

FIG. 8 .

 8 FIG. 8. Three Examples of Distributions of m(u) and n(u) In Equilibrium with: (a) f+(k) = 1.2, {b) f+(k) = 1.3, and (c) r(k) = 1.4415 The value of parameter a may be determined by enforcing the continuity of m and m' at the connecting point between m + and m-( 3!" -4 ) a =r sr-12 This distribution respects the strength condition <I a I and I m+ I ::5 1) if and only if 1.2 ::5 f"( k) :S 1.4415

FIG. 9 .

 9 FIG. 9. Failure Mechanisms Considered in Case of Simply Supported or Clamped Bottom Edge: (a) One Hinge Circle; (b) 1\vo Hinge Circles; (c) Three Hinge Circles -1 0

  APPENDIX I. RESULTS FOR CYL INDER WITH SUPPORTED EDGEIf k :S 0.01276, we have the following results:where u 1 and u 2 denote loci of the two hinges. The distribution mb must be completed in the upper part by two kinds of dis tributionsm, = � [cos ( 1T � = :: ) + 1 J (66a) m = .!. [ u-a) u2 -f u 3 +.!. (2a-f) u-.!. (3a -n]It can be proved that the distribution m, gives a static solution for 1 :Sf ::s; 1.759 (0 ::s; k ::s; 0.009061) and that m P gives a static solution for 1.421 ::s; f ::s; 1.893 (0.002371 ::s; k ::s; 0.01276).If 0.01276 ::s; k ::s; 0.07407, we have the following results with m�o the moment distribution, in the lower part (u :S u, ) and m2 for the upper part 1 k--18/2

2 (68)

If k =::: 0.7407, we have the following results:

1 ( u3 u ) f = 3; m = k -2 + u2 -2

APPENDIX II. RESULTS FOR CYLINDER WITH CL AMPED EDGE

If k :5 0.01108, we have the following results:

The distribution mb must be completed in the upper part of the shell by two kinds of distributions m. = � [cos ( 1r � = ::

It can be proved that m. gives a static solution for 1 :5 f :5 1.8762 (0 :5 k :5 0.00796) and that m P gives a static solution for 1.486 :5 f :5 2.0315 (0.002167 :5 k :5 0.01108).

If 0.01108 :5 k :5 0.05927, it can be proved that k is the only positive root of the following equation:

u, = ± �� ( f + 3) -16k (76c)

If k =::: 0.059217, the study has been completely carried out by [START_REF] Olszak | Die grenztragfahigkeit von zylin drischen schalen bei verscheidenen formen der plastizitatsbedingung[END_REF]. The results are


Uncertainties in Mean River Discharge Estimates Associated With Satellite Altimeter Temporal Sampling Intervals: A Case Study for the

  L'archive ouverte pluridisciplinaire

Uncertainties in mean river discharge estimates associated with satellite altimeter temporal sampling intervals: a case study for the annual peak flow in the context of the future SWOT hydrology mission.

F. Papa (1,*) , S. Biancamaria (1) , C. Lion (1) , W. B. Rossow (2) (1) LEGOS/CNRS-CNES-IRD-UPS, Toulouse, France (2) NOAA-CREST, City College of New York, New York, USA

Introduction

Continental freshwater runoff or discharge, as well as the spatial distribution and storage of fresh water on land, is a key parameter of the global water cycle and play an important role in driving the climate system [START_REF] Alsdorf | Measuring surface water from space[END_REF]. Moreover, natural disasters of hydrological origin dramatically affect human societies, with large economic losses during water-related extreme events such as floods or droughts.

Despite a widespread recognition of the need for better observations at global scale, surface freshwater measurements are still limited mostly to sparse in situ networks of gauges, the number of which has dramatically decreased during the last two decades, especially in remote areas [START_REF] Alsdorf | Measuring surface water from space[END_REF]. In addition, public access to recent observations is generally restricted.

Over the last twenty years, satellite remote sensing techniques have become more useful for hydrologic investigations [START_REF] Alsdorf | Measuring surface water from space[END_REF], [START_REF] Crétaux | Evolution of sea level of the big Aral Sea from satellite altimetry and its implications for water balance[END_REF], [START_REF] Papa | Interannual variability of surface water extent at global scale[END_REF]. In particular, satellite altimetry (TOPEX-Poseidon (T-P), Jason-2, ERS-1/2, GFO and ENVISAT missions) has been used for systematic monitoring of water levels in large rivers, lakes and floodplains [START_REF] Frappart | Interannual variations of river water storage from a multiple satellite approach: A case study for the Rio Negro River basin[END_REF] and several studies have demonstrated the capability of using these sensors locally for estimating river discharge in large rivers (still limited to rivers with a width of few kilometers), including the Ganges-Brahmaputra [START_REF] Papa | Satellite altimeter-derived monthly discharge of the Ganga-Brahmaputra River and its seasonal to interannual variations from 1993 to 2008[END_REF] or the Ob River [START_REF] Kouraev | Ob' River discharge from Topex-Poseidon satellite altimetry (1992-2002)[END_REF]. Indeed, the construction of empirical regression curves between altimetry-derived river water heights in large river basins and in situ measurements of river discharge can provide altimetry-based discharge estimates for times when in situ discharge observations are missing, or even, to extend the time-series of river discharge forwards/backwards. This technique has several limitations [START_REF] Alsdorf | Measuring surface water from space[END_REF], [START_REF] Papa | Satellite altimeter-derived monthly discharge of the Ganga-Brahmaputra River and its seasonal to interannual variations from 1993 to 2008[END_REF], [START_REF] Biancamaria | Preliminary Characterization of SWOT Hydrology Error Budget and Global Capabilities[END_REF], such as, to name a few, the quality of the current altimetry data themselves over continental water bodies, the current altimeter sampling frequency along track, and the spatial coverage of current satellite altimetry missions which is not adequate for global scale investigations due to their orbit track separation at the equator (few tens to hundreds of kilometers). In addition, a major drawback in the use of current altimetric measurements to monitor river stage and discharge is the temporal sampling rate at a given location, which is 10 days for T-P/Jason-2 and 35 days for ERS-1/2 and ENVISAT. With such space/time sampling intervals, current satellite altimeters cannot compete with observations made daily or twice daily by in situ gauges, a frequency required to study local hydrological processes, to evaluate flood risk or for the management of water resources. Nevertheless, for studies related to climate, the use of current radar altimetry is still extremely valuable as a complement to ground-based observations [START_REF] Papa | Satellite altimeter-derived monthly discharge of the Ganga-Brahmaputra River and its seasonal to interannual variations from 1993 to 2008[END_REF].

The future wide swath altimetry measurements made by the Surface Water and Ocean Topography (SWOT) satellite mission (to be launched in 2020) will provide high-resolution characterization of water surface elevations with 2D global maps of terrestrial surface water extent and storage changes and discharge estimates [START_REF] Alsdorf | Measuring surface water from space[END_REF]. Previous studies [START_REF] Alsdorf | Measuring surface water from space[END_REF], [START_REF] Biancamaria | Preliminary Characterization of SWOT Hydrology Error Budget and Global Capabilities[END_REF], [START_REF] Andreadis | Prospects for river discharge and depth estimation through assimilation of swath-altimetry into a raster-based hydrodynamics model[END_REF], [START_REF] Durand | Estimating river depth from remote sensing swath interferometry measurements of river height, slope, and width[END_REF] have reviewed the expected accuracy of the variables that will be measured and investigated the different errors which will affect SWOT data and the derived discharge. Instantaneous discharge estimated globally from SWOT are expected to have errors below 25%, even if locally these errors might be higher for ungauged basins. These errors are primarily due to errors on SWOT measurements (error on water elevation will be equal or below 10 cm, error on the water mask will be around 20% of the true area and error on the river slope will be equal or below 1cm/1km). Other anticipated sources of error come from ancillary data needed to compute the discharge (bathymetry and friction coefficient). When estimating monthly or average river discharge from instantaneous discharge estimates, the temporal resolution of the satellite observations will also be a source of uncertainties.

In this paper, we focus on the evaluation of the uncertainties due to the temporal sampling on the estimation of mean river discharge around the annual peak flow. Even though it is important to accurately monitor low flow and high flow discharge, the hydrologic events around the yearly peak flow are of particular importance as they are generally associated with the flood waves.

The in situ measurement and observation of river discharge are in general well established and, ideally, the goal for in situ discharge data accuracy is within +-5% of the true value.

However, given the difficulties to measure the depth and velocities (and consequently the true discharge), especially in large and strong-flowing rivers, the community agrees that a 15-20% accuracy is generally acceptable. When using radar altimetry, the accuracy of river discharge estimates depends, among other factors, on the satellite temporal sampling: for instance, mean discharge estimates will likely be more accurate for a river with several views per orbit than for a river with one revisit. Former and current radar altimeters (T/P, Jason-2, ERS-1/2, ENVISAT) view nadir along the orbit track, so a particular point is observed only once every repeat cycle except at overpasses (ascending and descending views) where two measurements are made. Given the inter-track interval (~300 km and ~80km at the equator respectively for T-P/Jason-2 and ERS-1/2 and ENVISAT) most of continental water bodies that are monitored are sampled only once and not always at an adequate location to measure river discharges. Unlike a nadir-viewing instrument, wide-swath instrument might see the same location from adjacent orbits, so a particular point might be observed several times every repeat cycle. With its wide swath altimetry measurements, the SWOT mission will offer a global spatial coverage with the number of views of a given location per cycle varying as function of latitude, and ranging from twice at the equator to more than 6 times at high latitudes. At the time of writing, SWOT nominal orbit will have a 22-day repeat period and a global coverage of the Earth up to the latitudes 78 o North and South.

Using T-P (10-day repeat cycle), ERS-2 and ENVISAT (35-day repeat cycle) altimeters, [START_REF] Papa | Satellite altimeter-derived monthly discharge of the Ganga-Brahmaputra River and its seasonal to interannual variations from 1993 to 2008[END_REF] and [START_REF] Kouraev | Ob' River discharge from Topex-Poseidon satellite altimetry (1992-2002)[END_REF] showed that the errors of the discharge estimated indirectly from altimetric measurements (at 10 days, monthly or annual time scales) are on average well within the range of acceptable errors (5 to 20%). However, the impact of the temporal sampling on the accuracy of the river discharge estimates during the annual peak flow is still not well known. For instance, if all overpasses occurred during flood stage, it leads to an overestimation of the average discharge based on these observations whereas in other cases, the sensor may completely miss the peak flow event. Over the Ganges-Brahmaputra river system, [START_REF] Papa | Satellite altimeter-derived monthly discharge of the Ganga-Brahmaputra River and its seasonal to interannual variations from 1993 to 2008[END_REF] showed that even with a coarse 35 day interval sampling interval (ERS-2/ENVISAT), the underestimation or overestimation of the in situ mean discharge in general never exceeds 20%.

Using a similar methodology, the goal of this study is to assess the effects that different altimeter repeat cycles (10-day, 22-day, 35-day temporal sampling and with a "real" SWOT repeat cycle in the wide-swath configuration) will have in estimating mean discharge around the yearly peak flow. For this we will use daily in situ gauge measurements from 11 large rivers around the world. Section 2 presents and discusses the datasets and the methodology. Section 3 presents and discusses the results. The conclusions are given in Section 4.

Dataset and methodology

We analyze the daily time series of in situ river discharge measurements for 11 large rivers (Table 1), which represent a fair sample of different environments, from the Tropics to Boreal regions. These time series are provided by 1) the HYBAM project (www.ore-hybam.org) for the Amazon and Congo Rivers and 2) the Global Runoff Data Center [START_REF]Long Term Mean Monthly Discharges and Annual Characteristics of GRDC Station/Global Runoff Data Centre[END_REF] for the other rivers. These 11 rivers were selected because of the availability of fairly long (more than a decade), accurate (evaluated) and continuous measurements.

Using these datasets, we performed the following analysis for each of the 11 rivers, with T representing the repeat cycle of the satellite (10, 22 or 35 days):

The date of the peak flow is identified for each year in the in situ record. A sliding window of T days is applied to the record, starting T days before the peak flow and going to the peak flow date in each year. The window moves with one-day steps; at each step, the average discharge is calculated using all T days in the window (the true mean) and using only the two end-points. The same calculation is done for all the years for which in situ discharge is available (Table 1). The difference (in percent of true mean) between the two means at each step is averaged over the years. The analysis is done for each of the 11 stations.

In parallel, "true" SWOT observation times were determined for each of the 11 gauge locations by calculating the number of times each gauge location is viewed from the satellite during a cycle using the relationship between the number of revisits and latitude [START_REF] Biancamaria | Preliminary Characterization of SWOT Hydrology Error Budget and Global Capabilities[END_REF]. The same analysis is done as previously but instead of considering only the 2-end points of the 22-day repeat cycle, we consider all observations of the target within the 22-day repeat cycle. The numbers of revisits per cycle for each station is given in Table 1, but note that the SWOT sampling is not uniformly distributed in time during one repeat cycle. Depending on the location, a target may be observed twice on two consecutive days and then not be sampled again for the next ten days. In our case, for the Amazon at Obidos, two observations are made on the 16 th and 17 th day of the cycle whereas there are up to six observations for the Lena River in Siberia, with irregular sampling on the 2 nd 5 th , 8 th , 11 th , 18th and 21 st days of the cycle.

Globally, the maximum time between two observations for a target is 13 days [START_REF] Biancamaria | Preliminary Characterization of SWOT Hydrology Error Budget and Global Capabilities[END_REF]. Note also, that in this study, we have only considered the SWOT measurements that observe the gauge location directly. However, because of its wide swath, SWOT will also measure water elevations upstream and downstream of the gauge location, which could then be used to infer water elevation at the gauge location using hydrodynamic models or statistical relationships and therefore, increase the number of samples on the mean discharge estimate [START_REF] Biancamaria | Assimilation of virtual wide swath altimetry to improve Arctic river modeling[END_REF]. Thus, the SWOT temporal sampling uncertainty computed in this study corresponds to the maximum expected error.

Finally, in the present study, we are interested in the effect of temporal sampling only. It is important to remind here that these uncertainties represent only a source of error among many other uncertainties associated with the estimates of instantaneous and mean river discharge from altimeter data. Indeed, as discussed in the introduction, the river water height needs to be first converted into discharge and such retrieval errors [START_REF] Biancamaria | Preliminary Characterization of SWOT Hydrology Error Budget and Global Capabilities[END_REF][START_REF] Durand | Estimating river depth from remote sensing swath interferometry measurements of river height, slope, and width[END_REF] will also largely impact the results. These effects will not be discussed here.

Results and Discussion.

The results for the 35-day, 22-day and 10-day temporal sampling are plotted in Figure 1 for the 11 stations. The x-axis values represent the lower endpoint of a T day sliding window. For 0, the lower end-point of the time window is at peak minus T days and the upper end-point is on the day of the peak discharge. At 5, the lower end is at peak minus T plus 5 days and the upper end is at peak plus 5 days and so on. The y-axis represents for each step the average difference over the years between the average discharge calculated using only the two endpoints and the true mean discharge calculated using all days in the time window. The y-axis values are expressed in percent of the true mean.

As expected, with the 35-day window (Figure 1a, the case of ERS/ENVISAT altimeters), the uncertainties are the largest, with big differences from river to river. The largest differences are found for the three basins in Siberia, the Ob, the Yenisey and the Lena, for which river discharge is characterized by a sharp and rapid increase at the end of the snowmelt season when the river ice breaks up. High river discharge values last only few weeks before a sharp decrease. For instance, for the Lena River, when one of the end-points is within +-5 days of the date of the peak flow, the average overestimation can be more than 200% of the true mean.

When the two samples bracket the peak flow date (around day 14 to 25), the underestimation is between 30 to more than 50%. The Yenisey and the Ob Rivers show the same patterns, but with smaller over/underestimation, especially for the Ob River for which the flood season and high peak flow last longer [START_REF] Papa | Ob' River flood inundations from satellite observations: A relationship with winter snow parameters and river runoff[END_REF][START_REF] Papa | Monitoring flood and discharge variations in the large Siberian rivers from a multi-satellite technique[END_REF]. For mid-latitude and tropical watersheds, the results show differences within the acceptable range of uncertainties for river discharges, i.e. around +-20%.

In most tropical watersheds (Amazon, Niger, Orinoco…), when one of the end-points is on the date of the peak flow (day=0 for instance), the mean discharge using the two end-points overestimates the true 35-day mean river discharge by about 10%. Then the differences show almost permanent underestimations of the 35-day mean discharge as soon as the peak flow is missed by 3 to 4 days. Moving the window forward shows that the differences (underestimates) are less than 5%, even with a 35-day sampling interval, and reach a maximum when the two samples bracket the peak flow date. The differences are larger for mid-latitudes watersheds, but the differences are generally less than 15% with a maximum underestimation of ~20% for the Mississippi. Note that among the tropical watersheds, the Irrawaddy shows the largest uncertainties (maximum and minimum underestimation above 20%) which might be explained by sharp increases and variability of the river discharge value during the monsoon season. This behavior is similar to the one found in [START_REF] Papa | Satellite altimeter-derived monthly discharge of the Ganga-Brahmaputra River and its seasonal to interannual variations from 1993 to 2008[END_REF] for two other large rivers of the region, the Ganges and the Brahmaputra.

As also expected, a ten-day sampling (Figure 1b), which is the repeat cycle of Topex-Poseidon and Jason-2 radar altimeters, leads to much smaller errors when estimating the discharge around the peak flow. Tropical basins, such as the Amazon, the Orinoco or the Congo, show almost no difference between the 10-day bracket and the true mean discharge (uncertainties below 2%). With a 10-day sampling interval, all rivers except the Lena are within +-20%. The maximum error for the Lena is an underestimation (25%) when the two samples bracket the peak flow date (day 5). Nevertheless, uncertainties for around day 0 for the Lena are reduced from more than 200% with a 35-day repeat cycle to ~20%. For the Yenisey and the Irrawaddy Rivers, the large uncertainties noticed with the 35-day sample are reduced to less than 10% with a 10-day repeat cycle.

Figure 1c gives the results for a 22-day repeat cycle for the SWOT mission with only nadir view, i.e. when the targets are visited only once every 22 days. As an intermediate case between the 35 and 10-day sampling intervals, the results still show fairly good estimates of mean discharge around the peak flow for most tropical basins (Amazon, Congo, Orinoco) and uncertainties in mid-latitude basins on the order of 10%. The Irrawaddy and Yenisey have largest errors but with maximum over/underestimations around 20%. For the Lena River, a sampling at twenty-two days is still too coarse to give an accurate estimate of the peak flow 0 mean discharge with uncertainties ranging from ~100% to -40%. However, as mentioned earlier, "true" SWOT observation times are more frequent per orbit repeat cycle with each gauge location sampled at least twice within a 22-day repeat cycle.

Using the real SWOT orbit sampling (Table 1), Figure 2 shows that the errors on estimated discharge around the peak flow are greatly reduced and well within the range of acceptable uncertainties for all 11 rivers. For the boreal and mid-latitudes basins (Figure 2a), the over/underestimation of mean discharges is always under 20%. The Lena, which is now sampled up to 6 times in a cycle also shows uncertainties within this range. For the Ob River, which already showed acceptable errors with a 22-day cycle (Figure 1c) is now sampled six times in a true SWOT configuration, reducing uncertainties less than 5%.

For the tropical watersheds (Figure 2b, 2 revisits minimum as in Table 1), all associated uncertainties are below 10%, except for the Irrawaddy, which still shows larger errors (overestimation of ~10% and underestimation of ~20%) even when it is sampled twice. For the Amazon, the Congo, the Mekong, the Orinoco, the uncertainties are on the order of a few percent. Thus, these results show that for the 11 rivers considered here, the uncertainties associated with SWOT temporal sampling when estimating mean discharge around the annual peak flow are well within the range of acceptable errors.

Absolute maximum mean discharge errors for each river (as in Figure 2a and2b) have been plotted as a function of the percentage of river discharge variance for frequencies above 1/(20 days) (Figure 3). This percentage is computed as follows. For daily each discharge time-series for the 11 rivers, a Fourier transform is calculated, and the integral of its variance (which is the square of the Fourier transform amplitude) over all time scales less than 20 days computed as a percentage of the total variance. This percentage gives the relative contribution of frequencies above 1/(20 days) to the discharge variance, and is expected to be larger for rivers with significant variability at shorter time scale. Figure 3 shows that the temporal sampling error is 1 associated with the short-time-scale variability of the river discharge time series. A regression analysis gives a quantitatively estimate of the relationship between discharge variance and SWOT temporal sampling errors in a form of a power law, statistically significant at 99% confidence level (R 2 =0.87 with 11 points, p-values<0.01 with |R|>0.735). Quite logically, for rivers with large short-time-scale variance, typically the boreal rivers with freeze/thaw cycles and the monsoon-affected Irrawaddy, the SWOT sampling error is larger. Usually, temporal sampling errors on mean river discharge are parameterized as a function of the river catchments area [START_REF] Biancamaria | Preliminary Characterization of SWOT Hydrology Error Budget and Global Capabilities[END_REF]. However, we show that in the case of estimates of the mean discharge around the seasonal peak flow, the uncertainty has a strong relationship with the variance of the river discharge. In the case of these 11 large rivers, the correlation between the absolute discharge errors and the catchment's area is only R 2 =0.18. Thus, the magnitude of the short-time-scale variance is a stronger predictor of the peak discharge error than the basin area. Although this analysis only had 11 samples, we suggest that the relationship with the variance could then be a new tool to infer the quality of future SWOT measurements at other gauge locations, if some past discharge time series is available to calibrate the relationship.

Conclusion.

This study reports a first effort to evaluate the uncertainties in the estimation of mean river discharge around the seasonal peak flow due to satellite altimeters temporal sampling intervals.

Analyzing the daily time series of in situ river discharge measurements for 11 large rivers in different environments, the results show that for high latitudes rivers, a sampling of 35 or 22 days in the nadir-looking configuration of current altimeter mission is too coarse to give an accurate estimate of the average discharge around the seasonal peak flow. For tropical watersheds however, such time sampling intervals lead to uncertainties that generally never exceed 20% and thus are in the range of uncertainties acceptable for river discharge estimations. On the other hand, the uncertainties associated with a 10-day repeat cycle are well within the range of acceptable errors from Tropical to Siberian rivers. Thanks to its wide swath altimetry technique, which will enable to observe a given location at least twice at the equator and up to six times in high latitudes every repeat cycle, the uncertainty due to SWOT time sampling on the average discharge around the seasonal peak flow is greatly reduced when compared to a 22-day repeat cycle instrument with a nadir looking angle. We found that these uncertainties are generally well within the range of acceptable errors for boreal watersheds (absolute maximum mean discharge uncertainties from 5 to 20%), mid-latitudes watersheds (absolute maximum mean discharge uncertainties ~10%) and tropical watersheds (absolute maximum mean discharge uncertainties from 2 to ~20%). Moreover, we find that absolute maximum mean discharge uncertainties around the seasonal peak flow have a strong relationship with the variance of the river discharge. Thus, around the peak flow, we suggest that the magnitude of the short-time-scale variance of the discharge could be used as predictor of the uncertainties rather than the commonly used basin area.

The future launch of the SWOT mission in 2020 will represent a step increase for continental hydrology and further studies are needed to refine the SWOT error budget for discharge estimates. For instance, the uncertainties for smaller rivers (~100 m to ~1 km width) have not been addressed here and require further investigations. Moreover, we address in this study the source of errors due to the temporal sampling of the satellite only, but in reality it will combined with other sources of uncertainty. These issues need to be addressed in future works.

Figures Caption:Figure 1 :

 1 Figures Caption:

Figure 2 :Figure 3 :

 23 Figure 2: Same as Figure 1 with the SWOT 22-day repeat cycle but taking into account the

  Figures:

Figure 3 .

 3 Figure 3.

Table 1 :

 1 Information on the daily in situ river discharge time series used in this study: River name, Gauge station name and location, first and last year of the available time series, catchment's area, mean value and standard deviation (STD) for the entire daily discharge time series and the number of views of the a given gauge station per SWOT cycle.

	River	Station Name and locations	Time series Catchment area	Mean Discharge/	Number of
	Name			(km 2 )	STD (m 3 /s)	samples per
						SWOT cycle
	Amazon	Obidos (1.92 o S; 55.67 o W)	1968-2008	4618000	172700 / 49840	2
	Congo	Brazzaville (4.25 o S; 15.28 o E)	1968-2008	3500000	40500 / 9300	2


Hui-Jia Cao

Vincent Perrichot
email: vincent.perrichot@univ-rennes1.fr

Chungkun Shih

Dong Ren

Tai-Ping Gao
email: tpgao@cnu.edu.cn

A revision of Haidomyrmex cerberus Dlussky (Hymenoptera: Formicidae: Sphecomyrminae) from mid-Cretaceous Burmese amber

Keywords:

The type species of the genus Haidomyrmex Dlussky, 1996 is revised. Haidomyrmex cerberus was originally described based on a partial worker specimen only, which was later re-figured with some erroneous characters. Two worker specimens assignable to this species were recently discovered in the collection of Burmese amber from the Capital Normal University of Beijing. A revised description and comparison with the two other known species of Haidomyrmex are provided, and the diagnoses of the genus and species are emended.

Introduction

Fossils are unique records of the past diversity of life, providing the only evidence on stem groups or extinct lineages that vanished millions of years ago. Some fossils that are exquisitely preserved can provide a myriad of morphological and ecological data on an extinct taxon. But there are a lot of species known and described from a single fossil individual that is incompletely preserved, thus remaining only partially informative. In 1996, Dlussky described Haidomyrmex cerberus, the first ant species ever found in Burmese amber and a highly intriguing ant with a bizarre cranio-mandibular morphology. Unfortunately, the type and unique specimen was only partially preserved, missing parts of the antennae, legs and gaster. Some characters, especially those of the head, were also obscured due to the turbidity of the amber piece and the position in which the specimen was exposed. Still, Dlussky provided a rather detailed description including some hardly visible features, and he suggested this ant to be analogous to the modern trap-jaw ants, but placed the genus in the extinct Sphecomyrminae [START_REF] Dlussky | Ants (Hymenoptera: Formicidae) from Burmese amber[END_REF]. This placement was followed by [START_REF] Bolton | Synopsis and classification of Formicidae[END_REF] who erected a new tribe Haidomyrmecini for this genus. Engel and Grimaldi (2005: figs. 7-8), however, proposed a re-interpretation of the holotype and modified the generic diagnosis. In contrast to the features described by [START_REF] Dlussky | Ants (Hymenoptera: Formicidae) from Burmese amber[END_REF], [START_REF] Engel | Primitive new ants in Cretaceous amber from Myanmar, New Jersey, and Canada (Hymenoptera: Formicidae)[END_REF] could not see the inner tooth on the mandibles, the long trigger setae on the clypeus, the stiff seta on the third antennomere, the remnant ocelli, and the subpetiolar process. They also suggested that the mandibles were not preserved in their natural position but significantly distorted, and erroneously noted similarities with the genus Brownimecia [START_REF] Grimaldi | New and rediscovered primitive ants (Hymenoptera: Formicidae) in Cretaceous amber from New Jersey, and their phylogenetic relationships[END_REF], from New Jersey amber. But the discovery of Haidomyrmodes [START_REF] Perrichot | New fossil ants in French Cretaceous amber (Hymenoptera: Formicidae)[END_REF], a genus closely similar to Haidomyrmex, later revealed that [START_REF] Dlussky | Ants (Hymenoptera: Formicidae) from Burmese amber[END_REF] had correctly interpreted the mandibles [START_REF] Perrichot | New fossil ants in French Cretaceous amber (Hymenoptera: Formicidae)[END_REF]. The haidomyrmecines' bauplan was further confirmed by the subsequent discoveries of the genera Haidoterminus [START_REF] Mckellar | A new trap-jawed ant (Hymenoptera: Formicidae: Haidomyrmecini) from Canadian Late Cretaceous amber[END_REF], Ceratomyrmex Perrichot, Wang & Engel, 2016, and Linguamyrmex Barden & Grimaldi, 2017[START_REF] Mckellar | A new trap-jawed ant (Hymenoptera: Formicidae: Haidomyrmecini) from Canadian Late Cretaceous amber[END_REF][START_REF] Perrichot | Extreme morphogenesis and ecological specialization among Cretaceous basal ants[END_REF][START_REF] Barden | A new genus of hell ants from the Cretaceous (Hymenoptera: Formicidae: Haidomyrmecini) with a novel head structure[END_REF]. But some confusion remains on the strict definition of Haidomyrmex and the Haidomyrmecini, since two additional species of Haidomyrmex were described with 12 antennal segments while depicted with only 11 segments (Barden and Grimaldi, 2012: fig. 4), and the presence of the trigger setae, reduced ocelli and inner mandibular tooth was confirmed on the holotype of H. cerberus (Perrichot et al., 2016: supplemental information).

Finally, despite the discovery of several hundreds of additional specimens of Haidomyrmex in Burmese amber during the last decade, it is striking that none could be assigned to H. cerberus again (Perrichot, pers. data from various institutional and private collections). Instead, they largely belong to H. scimitarus [START_REF] Barden | Rediscovery of the bizarre Cretaceous ant Haidomyrmex Dlussky (Hymenoptera: Formicidae), with two new species[END_REF], more rarely to H. zigrasi [START_REF] Barden | Rediscovery of the bizarre Cretaceous ant Haidomyrmex Dlussky (Hymenoptera: Formicidae), with two new species[END_REF], the two other species described more recently [START_REF] Barden | Rediscovery of the bizarre Cretaceous ant Haidomyrmex Dlussky (Hymenoptera: Formicidae), with two new species[END_REF]. A reason for this could be that the type specimen of H. cerberus, although described in 1996, originated from an early collection of Burmese amber that was sent to Cockerell by Swinhoe in 1920 [START_REF] Ross | A list of type and figured specimens of insects and other inclusions in Burmese amber[END_REF][START_REF] Zherikhin | A review of the history, geology and age of Burmese amber (Burmite)[END_REF]. Thus, the holotype might have been collected from a locality and/or geological stratum that was potentially distinct from the current localities from where Burmese amber were mined for the last decade.

However, in a study of 250 ants from the Burmese amber collection that was recently established at the Capital Normal University of Beijing, two out of nine Haidomyrmex are assignable to H. cerberus. This discovery allows for a complemental description of the species and emendations for the generic and specific diagnoses.

Material and methods

Depository, horizon, and specimen handling

This study is based on new amber material from Hukawng Valley in Kachin State of northern Myanmar located at the north end of Noije Bum, at 26°15′N, 96°34′E, some 18 km south-west of the town of Tanai. This deposit is dated to 98.79 ± 0.62 Mya based on radiometric uranium-lead dating [START_REF] Shi | Age constraint on Burmese amber based on U-Pb dating of zircons[END_REF], and the recent finding of an ammonite embedded in amber and assignable to Puzosia (Bhimaites) supports a Late Albian-Early Cenomanian age of the amber [START_REF] Yu | An ammonite trapped in Burmese amber[END_REF]. The amber is housed in the Key Lab of Insect Evolution & Environmental Changes, Capital Normal University, Beijing, China. (CNUB; Dong Ren, Curator) Specimen No. CNU-HYM-MA2019051 is complete, without apparent distortion, and preserved in a piece of yellow amber containing a suspension of bubbles and organic fragments. Specimen No. CNU-HYM-MA2019052 is almost complete, missing only the left antenna beyond scape and left foreleg beyond femur, and is only weakly distorted. It is preserved in a piece of clear yellow amber with stellate hairs, fragments of a spider web and organic debris, and has tiny bubbles or dust covering the cuticle in places. Both amber pieces were ground and polished to form small cabochons prior to acquisition, but a further polishing in the CNU Lab was made using silicon carbide papers for us to inspect the specimens in optimal views.

The specimen CNU-HYM-MA2019051 was examined and photographed using a Nikon SMZ 25 microscope equipped with a Nikon DS-Ri 2 digital camera system.

The specimen CNU-HYM-MA2019052 was examined and photographed using a Leica MZ8 stereomicroscope equipped with a Canon 5D Mark II digital camera, and stacks of photographs taken at different focal planes were merged using Helicon Focus software. The line drawing and figures were produced using Adobe CC (Illustrator and Photoshop). Measurements were obtained using the measurement tool of Nikon software.

The type specimen NHM-In.20182 was examined and photographed at the Sackler Lab of the Natural History Museum of London in 2013. The photographs are available on AntWeb (at www.antweb.org).

Systematic palaeontology

Family Formicidae Latreille, 1809 Subfamily Sphecomyrminae Wilson & Brown, 1967Tribe Haidomyrmecini Bolton, 2003 Genus Haidomyrmex [START_REF] Dlussky | Ants (Hymenoptera: Formicidae) from Burmese amber[END_REF] Type species. Haidomyrmex cerberus Dlussky, 1996: 451, 2.2 times length of pedicel; lengths of antennomeres: scape 0.53, pedicel 0.12; flagellomeres F1-F10: 0.16, 0.23, 0.22, 0.20, 0.20, 0.22, 0.18, 0.17, 0.16, 0.21; apex of scape slightly broadened, its ventral margin bearing short erect setae; FI with stiff, erect seta on ventral surface (Fig. 2B). Clypeal process a small hemispheric lobe (maximal diameter 0.12) moderately elevated, dorsal surface coated by brush of stiff spine-like setae gradually longer and thicker toward ventral margin of lobe, ventral surface with one visible pair of trigger setae [two pairs visible] flanking tips of mandibles; trigger seta length 0.31. Anterior clypeal margin concave, bearing 4 erect stiff setae (Fig. 2C). Mandibles (Fig. 2C) parallel and close to each other, basal portion 0.51 in length (ignoring slight curvature), curved apical portion 0.67 in length; curved portion gradually tapered to an acute point, apex nearly reaching to tip of clypeal process, ventral margin of curved portion coated in dense setae from base to terminal quarter; inner surface of mandibles concave, particularly from inner triangular blade to apical two thirds of curved portion; anterior margin of triangular blade with a row of 7-8 spine-like setae directed inward (Fig. 2C); triangular blades overlapping posteriorly, apparently symmetrical and each with a single minute tooth.

Maxillary palps (Fig. 2D) long, exposed length 0.53, with 5 visible articles (basal article mostly concealed between mandibles). Labial palps (Fig. 2D) short, exposed length 0.08, with only 2 apical articles visible (but palp formula should be 5:3 according to undescribed specimens of Haidomyrmex with palpomeres fully exposed; Perrichot, pers. data from various institutional and private collections).

Mesosoma. Long and slender; depth (greatest dorso-ventral distance) 0.45, length (including neck) 1.98. Neck narrow and long, pronounced in lateral view. Pronotum elongate; pronotal dorsal outline feebly convex in its anterior third, flat and sloped in its posterior two thirds (Fig. 1A,B; Fig. 2A). Mesosomal length 0.38; mesonotum convex (Fig. 1A,B), maximum width 0.12 and 0.08 in height. A distinct metanotal sclerite posteriorly to this bulging mesoscutellum. Propodeum longer than high, gradually sloping posteriorly; metapleural gland opening semicircular, facing posteroventrad; metapleural bulla developed, roughly hemispherical. Legs long and slender. Length of procoxa 0.65, mesocoxa 0.39, metacoxa 0.32. Length of meso-and metatrochanters 0.20. Length of pro-and mesofemur ca.1.00, metafemur 1.84. Length of protibia 0.86, mesotibia 0.91, metatibia 1.67. Protibia apically with one long pectinate spur (calcar) and two short simple spurs. Mesotibia with two simple spurs, metatibia with one large pectinate spur and one short simple spur (Fig. 1D). Several short and stiff setae also present apically on hind tibia. Total length of pro-, meso-, and metatarsi 1.10, 1.37, 1.90, respectively. Basal tarsomere longest, remaining four tarsomeres shortening gradually, length of metatarsal segments I-V: 0.87, 0.46, 0.30, 0.18, 0.20. Surface of tarsomeres covered with fine setae. Pretarsal claws with distinct subapical tooth, arolium well-developed.

Metasoma. Petiole (Fig. 1C) with short anterior peduncle, dorsal margin broadly rounded, strongly narrowed posteriorly; petiole length 0.58, maximal height 0.32. Subpetiolar process (Fig. 1C) 0.06 at greatest height, in profile a small triangle with posterior margin distinctly longer than anterior margin. Lateral sulcus visible running anteroposteriorly along petiole. First gastral segment with helcium pronounced, forming a narrow post-petiolar peduncle. Gaster 1.45 in length (excluding sting), with gastral segment I 0.48, segment II 0.57; sutures between tergite and sternite of segments I and III distinct; distal part of gastral segment V (pygidium and hypopygium) setulose. Sting long, robust; externalized part enclosed by gonostyli/third valvulae.

Locality and horizon. Hukawng Valley, Kachin State, northern Myanmar; formed near the Aptian-Cenomanian boundary [START_REF] Shi | Age constraint on Burmese amber based on U-Pb dating of zircons[END_REF], ca. 99 Mya.

Remarks. Assignment of these two specimens to H. cerberus is based on the similarities in the body sizes (4.5 to 5 mm, vs. 8 mm for H. scimitarus and 3.5 mm for H. zigrasi). The antennae are also similar, with the scape as long as the three following antennomeres combined (vs. scape distinctly longer in H. scimitarus and H. zigrasi), and with F1 ca. 1.2 times the length of pedicel, F2 ca. twice the length of pedicel (vs. F1 twice as long as pedicel, F2 4 times the length of pedicel in H. scimitarus; F1 slightly shorter than pedicel, F2 as long as pedicel in H. zigrasi).

Finally, the body covered by dense patches of erect setae, combined with the dense pubescence of the basal antennomeres and outer surface of the mandibles, are characteristic of H. cerberus, but absent in the other two species.

Conclusion

The present discovery confirms that all species of Haidomyrmex possess similar mandibles with the inner surface armed with a triangular blade, and paired trigger setae ventrally on the clypeus. It is likely that they all have the inner surface of mandibles concave and with a ventral row of stiff setae, although these characters could not be assessed for H. zigrasi since the only know specimen has the mandibles tightly closed. The specimen CNU-HYM-MA2019052 is preserved with the mandibles entirely aligned and their triangular blades overlapping, suggesting that the inner concavities form a tube-like channel. Similar concave mandibles forming a channel were observed on Linguamyrmex, another genus of haidomyrmecine ants that might have used this system to feed on liquid [START_REF] Barden | A new genus of hell ants from the Cretaceous (Hymenoptera: Formicidae: Haidomyrmecini) with a novel head structure[END_REF]. Some extant ants feed on honeydew or hemolymph [START_REF] Hölldobler | The Ants[END_REF][START_REF] Saux | Dracula ant phylogeny as inferred by nuclear 28S rDNA sequences and implications for ant systematics (Hymenoptera: Formicidae: Amblyoponinae)[END_REF]). It indicates that Haidomyrmex might have fed on liquid too, as maybe all members of the Haidomyrmecini, or so-called "hell ants". This tribe currently contains seven species within five genera of potentially highly specialized predators, and encompassing most of the Late Cretaceous: Haidomyrmex (3 sp.), Ceratomyrmex (1 sp.) and Linguamyrmex (1 sp.), from Burmese amber; Haidomyrmodes (1 sp.) from Albian-Cenomanian French amber; and Haidoterminus (1 sp.) from Campanian Canadian amber [START_REF] Perrichot | New fossil ants in French Cretaceous amber (Hymenoptera: Formicidae)[END_REF][START_REF] Perrichot | Extreme morphogenesis and ecological specialization among Cretaceous basal ants[END_REF][START_REF] Barden | Rediscovery of the bizarre Cretaceous ant Haidomyrmex Dlussky (Hymenoptera: Formicidae), with two new species[END_REF][START_REF] Mckellar | A new trap-jawed ant (Hymenoptera: Formicidae: Haidomyrmecini) from Canadian Late Cretaceous amber[END_REF][START_REF] Barden | A new genus of hell ants from the Cretaceous (Hymenoptera: Formicidae: Haidomyrmecini) with a novel head structure[END_REF]Barden, 2017: table. 1). Indeed, all female haidomyrmecines share long sickle to L-shaped mandibles that uniquely moved in a vertical plane aligned with the longitudinal axis of the body, and a clypeus elongate and posteriorly with a prominent setose pad (vs. short mandibles transversely aligned and moving laterally, and clypeus transverse and anteriorly setose in the remaining Sphecomyrminae).

Figure Captions

  Figs.1-3

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Haidomyrmex cerberus, new material CNU-HYM-MA2019051 from

Acknowledgments

We express our sincere gratitude to Ms. Xiaodan Lin and Ms. Yimo Wang (College of Life Sciences, Capital Normal University) for their practical help to the first author; and to Cédric Chény (Univ Rennes) for assistance in the imaging of one of the specimens. We also appreciate the valuable comments and constructive suggestion from Dr. Eduardo Koutsoukos, Prof. André Nel and two anonymous reviewers. D.R. was supported by grants from the National Natural Science Foundation of China (grant nos. 31730087, 41688103), the Program for Changjiang Scholars and Innovative Research Team in University (IRT-17R75), and Project of High-level Teachers in Beijing Municipal Universities (grant no. IDHT20180518); T.P.G. was supported by the Support Project of High-level Teachers in Beijing Municipal Universities in the Period of 13th Five-year Plan (CIT&TCD201704090) and Capacity Building for Sci-Tech Innovation -Fundamental Scientific Research Funds.

Examination and imaging of the type specimen at the NHM was made possible through a Synthesys visiting grant (#GB-TAF-2549 to V.P. in 2013). The authors declare that there is no competing interests in the manuscript.