Meritxell Fernández-Coll

Thomas Arbez

Federico Bernardini

Josep Fortuny

Cranial anatomy of the Early Triassic trematosaurine Angusaurus (Temnospondyli: Stereospondyli): 3D endocranial insights and phylogenetic implications

Keywords: Trematosauria, Trematosauroidea, Trematosauridae, Trematosaurinae, Stereospondyli Resumen Trematosauria, Trematosauroidea, Trematosauridae, Trematosaurinae, Stereospondyli

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Temnospondyls are a large group of tetrapods ranging from the Early Carboniferous to the late Early Cretaceous [START_REF] Schoch | The evolution of major temnospondyl clades: An inclusive phylogenetic analysis[END_REF]. They survived the end-Permian mass extinction and during the Early Triassic (Induan) their diversity rapidly increased, although it decreased gradually after the Late Triassic [START_REF] Milner | The radiations of temnospondyl amphibians[END_REF]). One of the most successful clades of temnospondyls are stereospondyls. Their members made their first appearance during the Late Permian and survived until the Early Cretaceous, reaching their maximum diversity during the Early Triassic [START_REF] Stayton | Geometric morphometrics of the skull roof of stereospondyls (Amphibia: Temnospondyli)[END_REF], presenting a worldwide distribution. They were predominantly, but not exclusively, aquatic-semiaquatic tetrapods living within a range of habitats from freshwater to coastal and even marine environments. During the beginning of the Early Triassic two stereospondyl groups, trematosauroids and capitosauroids, dominated these ecosystems, some acquiring gigantic sizes [START_REF] Fortuny | Comparative 3D analyses and palaeoecology of giant early amphibians (Temnospondyli: Stereospondyli)[END_REF]. Trematosauroids lived from the Early to the Late Triassic [START_REF] Hellrung | Revision von Hyperokynodon keuperinus Plieninger (Amphibia: Temnospondyli) aus dem Schilfsandstein von Heilbronn (Baden-WuÈrttemberg)[END_REF], expanded over Pangaea, and are known from all continents except South America and Antarctica. The phylogenetic relationships within trematosauroids are still debated, but they include few basal trematosauroids (such as Benthosuchus) and a huge clade of trematosaurs divided into two subclades: Trematosaurinae (short-snouted members) and Lonchorhynchinae (long-snouted members; see [START_REF] Welles | A review of the lonchorhynchine trematosaurs (Labyrinthodontia), and a description of a new genus and species from the Lower Moenkopi Formation of Arizona[END_REF][START_REF] Steyer | The first articulated trematosaur "amphibian" from the lower Triassic of Madagascar: Implications for the phylogeny of the group[END_REF][START_REF] Schoch | A complete trematosaurid amphibian from the Middle Triassic of Germany[END_REF][START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF] for further discussion).

Most trematosaurs belong to Trematosaurinae. Several trematosaurine taxa have been described, mainly from Central European basins and the Russian Eastern Platform, but also from other European basins (e.g. Spitsbergen), Africa (e.g. Madagascar, South Africa) and Australia. The radiation of trematosaurines during the Early Triassic was rapid, demonstrating the importance of this group in understanding the recovery of the ecosystems just after the End Permian mass extinction.

Herein, the Early Triassic trematosaurine Angusaurus is analyzed and its phylogenetic position addressed. This genus is known from the Russian East Platform and four species have been erected to date based on cranial and mandible specimens [START_REF] Kuzmin | Lower Triassic Stegocephalia from the northern part of the Oka-Tsna ridge. Trematosaurus weidenbaumi nov. sp[END_REF][START_REF] Getmanov | A labyrinthodont from the Lower Triassic of the Obshchiy Syrt region[END_REF][START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF][START_REF] Novikov | New Early Triassic labyrinthodonts of the Middle ForeUrals[END_REF]). Based on the description of an undescribed smallsized specimen our aims are: (a) to describe the external anatomy; (b) to describe in detail its neurocranium using MicroCT scanning; (c) the taxonomic study of the analyzed skull to evaluate its taxonomical assessment; and (d) to perform a cladistic analysis of trematosaurids, with particular emphasis on trematosaurines in order to evaluate the phylogenetic position of Angusaurus and shed new light on the phylogeny of trematosaurines and its paleobiogeographical implications.

Institutional Abbreviations-MNHN: Museum National d'Histoire Naturelle, Paris, France; PIN: Paleontological Institute, Moscow, Russian Federation; SMNS, Staatliches Museum für Naturkunde in Stuttgart, Germany.

Materials and methods

The analyzed specimen (SMNS 81782) is stored at the Staatliches Museum für Naturkunde in Stuttgart, Germany and corresponds to an almost complete skull. It has been reported (not figured nor described) by [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF] and referred to A. succedaneus. The specimen was recovered in the region of Rybinsk, Russia. This area is well-known geologically and paleontologically, located in the upper part of the Volga River. The sediments consists of lacustrine marls dated as Olenekian, Early Triassic (Grauvogel-Stamm and Ash 2005). The region of the upper Vetluga River, one of the Volga effluents, is well known for the presence of temnospondyl remains: capitosaurs such as Wetlugasaurus, trematosauroids such as Benthosuchus and trematosaurids as Thoosuchus, Angusaurus and Trematotegmen, have all been found in the Upper Vetluga formations [START_REF] Sennikov | Evolution of the Permian and Triassic tetrapod communities of Eastern Europe[END_REF].

The specimen was scanned using X-ray computed microtomography (microCT) at the Multidisciplinary Lab (MLAB) of the "Abdus Salam" International Centre for Theoretical Physics (ICTP) in Trieste, Italy. See [START_REF] Tuniz | The ICTP-Elettra X-ray laboratory for cultural heritage and archaeology[END_REF] for further details about the equipment. Due to the size of the sample and the reduced size of the microCT scan detector, it was required to scan it in two parts: one from the anterior end of the snout to the posterior edge of the prefrontal, and the second one from the posterior edge of the prefrontal to the occipital condyles. The parameters for each part of the skull were the same: 149 kV voltage, 50 μA current with 0.1 mm filter of copper. A total of 1800 X-ray slices were obtained with a voxel resolution of 35.71 μm.

Raw data from each scan was imported (as stack of TIFF 8-bit files) to Avizo 7.0 to generate a 3D surface from the microCT images. Data produced by segmentation in Avizo were exported to the programs Rhinoceros 5.0 and 3matic 9.0 for PDF 3-D creation (see supplementary information).

The direct study of the specimen complements the digital part of its analysis. Measurements of some areas of the skull were taken with a digital calliper (Electro DH, model 60.205, resolution of 0.01 mm).

Anatomical abbreviations ans, anterior sulcus; apv, anterior palatal vacuity; br, basipterygoid ramus; ch, choana; cls, cristae laterosphenoidales; co, crista oblica; cp, cultriform process; ct, chorda tympani; cv, columna verticalis; f, foramen; fm, foramen magnum; is, infraorbital sulcus; iv, interpterygoid vacuity; js, jugal sulcus; jf, jugular foramen; la, lamina ascendens; ld, lamina descendens; fl, flexura lacrimalis; lp, lamellar process; mf, Meckelian foramen; na, naris; oc, occipital condyle; ocs, occipital sulcus; pap, paroccipital process; pf, pineal foramen; pl, processus lamellosus; plf, palatoquadrate fissure; pr, palatine ramus; psm, processus submedullaris; pst, processus subtympanicus; ptf, postemporal fenestra; qr, quadrate ramus; sos, supraorbital sulcus; sv, supratemporal vacuity; ts, temporal sulcus. Bones: AN, angular; COR, coronoid; D, dentary; ECPT, ectopterygoid; EO, exoccipital; F, frontal; ICOR, intercoronoid; J, jugal; L, lacrimal; MX, maxilla; N, nasal; P, parietal; PAR, prearticular; PCOR, precoronoid; PL, palatine; PMX, premaxilla; PO, postorbital; POF, postfrontal; PP, postparietal; PRF, prefrontal; PS, parasphenoid; PSP, postsplenial; PT, pterygoid; Q, quadrate; QJ, quadratojugal; SA, surangular; SP, splenial SQ, squamosal; ST, supratemporal; T, tabular; VO, vomer.

Systematic paleontology

Description

Preservation

The specimen (SMNS 81782) has a narrow, elongated and triangular-shaped skull (Fig. 1). Its maximum length is 112 mm and it has a maximum width of 74 mm ). The specimen is a nearly complete and well-preserved skull (Figs. 1,2). The anteriormost part of the snout, formed by the maxilla, premaxilla and part of the nasal bone, as well as the cheek region, comprising the jugal and the quadratojugal, is the only slightly damaged part of the specimen that precludes proper observation. There is no evident damage due to mechanical transport in the skull. The posterior margin of the skull, formed by the tabulars and the postparietals, is slightly concave (Figs. 1,2).

Lateral line system and ornamentation

Between most of the bones the sutures are clearly visible, while on the anterior part of the snout-as previously mentioned-and on the cheek region they are less well defined due to damage. Also, on some bones (nasal, premaxilla, maxilla) the ornamentation and the sensorial sulci are difficult to discern with confidence. Sensorial sulci are generally continuous and strongly marked (Figs. [START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]2). The anterior sulcus, generally found at the anterior end of the snout, close to the naris, is faintly present. The supraorbital sulcus is continuous with an S-shaped morphology and passes through the nasal, lacrimal, prefrontal, frontal, postfrontal, and postorbital bones, where it connects with the jugal and the temporal sulci, the latter found on the left side (but not present on the right side). The infraorbital sulcus goes mainly through the maxilla, and in the lacrimal bone it forms a flexure, the flexura lacrimalis, that is faintly marked and rounded. The jugal sulcus is continuous and goes through the jugal, following the quadratojugal-squamosal suture in posterior direction (Fig. 2). The temporal sulcus contacts the jugal sulcus in the jugal and goes through the postorbitals and the supratemporals, contacting the supraorbital sulcus in the postorbital bone (clearly on the left side and almost contacting it on the right side). The occipital sulcus is faintly present on the left side, from the tabular horn to up to one-third of the postparietal. On the right side, the occipital sulcus is only visible on the tabular bone. On the right side of the skull roof, part of the supraorbital, temporal and occipital sulci are slightly marked. On this side, the supraorbital sulcus is less evident through the postfrontal while the temporal sulcus is less evident through the supratemporal. On the other hand, the occipital sulcus its less evident on the postparietal bone.

The skull roof has a prominent ornamentation with some areas presenting ridge-groove ornamentation, indicating intensive growth on the skull [START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF]. The bones on the skull roof are characterized by dermal bone with polygonal sculpture in the ossification centre and radial sculpture in the peripheral region of the bone. The polygonal sculpture consists of polygonal grooves separated by thin ridges, while the radial sculpture is characterized by elongated furrows and ridges that radiate in opposite direction from the ossification centre of the bone [START_REF] Witzmann | Sculpture and vascularization of dermal bones, and the implications for the physiology of basal tetrapods[END_REF]). The sculpture is not homogenous along the entire skull roof. Suture morphology was identified thanks to micro-CT images. Most sutures are groove-like (sensu

Skull roof

The premaxilla and the maxilla are anteroposteriorly elongated but badly preserved anteriorly, precluding assessment of the morphology of the anteriormost part of the snout. The nasal bones are more elongated than the premaxilla. The premaxilla-nasal suture morphology cannot be discerned due to damage. The nostrils, situated at the tip of the snout, are oval, laterally positioned, and have a length between 8 and 9 mm. This measurement is approximate because the left one is badly preserved. The nostril borders are formed by the premaxilla, the nasal and the maxilla. There is no trace of a septomaxilla, although its presence cannot be completely discarded due to bad preservation. The lacrimal is present, with the supraorbital and infraorbital sulci passing through it. The latter sulcus includes the flexura lacrimalis. The prefrontal connects with the postfrontal, frontal, nasal, lacrimal and jugal bones. The prefrontal is longer than wide, being wider in its posterior part. The jugal is long, forming part of the orbital rim, and representing most of the orbital margin (35%). The rest of bones that form the orbital margin (postorbital, postfrontal, prefrontal, and jugal) occupy a slightly smaller part of the contour, around 20% each (Fig. 1). The frontal is one of the longest bones of the skull roof with a maximum length of 40.1 mm (total length of the skull is 112 mm). It is followed by the nasal, which is approximately 38.3 mm in length. The frontal does not contact the orbital margin, a characteristic found in other stereospondyls.

The orbits are oval and large in size, 19.1 mm long (right) and almost 18 mm long (left), around 16.6% of the total preserved dorsal midline length. The orbits are placed in the posterior half of the skull and are lateralized, located near the edge of the skull. The interorbital distance is 23.3 mm (supplementary material-Table ). There is a pronounced depression between the two orbits, formed by part of the postfrontal and frontal bones. The ventral opening of the orbits is placed on the posterior half of the interpterygoid vacuities. The shape of the postorbital bone is raindrop-like, being posteriorly elongated. The supratemporal is rectangular and contacts the tabular, postparietal, parietal, postfrontal, postorbital and squamosal bones. The postfrontal is also rectangular. The quadratojugal is elongate and narrower than its adjacent bones. The quadrate condyles are anterior to the exoccipital condyles. The squamosal is longer than wide and articulates with the supratemporal, postorbital and quadratojugal bones. The parietal is elongate and contacts the postparietal, supratemporal, postfrontal and frontal. The pineal foramen is circular and large, with a diameter of almost 4 mm. It is situated in a posterior position between both parietals, 8.7 mm posterior to the orbits (supplementary material-Table ). The tabular is triangular, with a short, pointed and poorly-developed horn (Figs. 1,2). The tabular and squamosal bones conform an open and deep otic notch. The postparietal is slightly wider than long.

Palate

In ventral view, the palatal region of Angusaurus is similar to Thoosuchus [START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF], although some differences deserve special attention (Figs. 1,2). The premaxilla-vomer suture possibly reaches the anterior margin of the anteropalatal vacuity (or fenestra sensu [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF], or ends really close to it. These vacuities (Fig. 1) are rounded and have a diameter of 5 mm (right) and 4.9 mm (left). The vomer-palatine suture is located on the posterior part of the choana. The choanae are oval, both 11.9 mm on the long axis, and placed on the lateral margin of the snout. The maxilla is very elongated and contacts the vomer, palatine, ectopterygoid and jugal bones. The palatine contacts the vomer, ectopterygoid, and maxilla. The edge of the palatine forms part of the interpterygoid vacuity and the choana (Figs. 1,2). The palatine is adjacent to the ectopterygoid posteriorly. The ectopterygoid bears some dentition, mainly anteriorly and on the central region. The ectopterygoid contacts the maxilla laterally, the pterygoid posteriomedially, and the jugal dorsolaterally. The ectopterygoid forms the central part of the margin of the interpterygoid vacuities.

The pterygoid is tri-radiate, with a palatine, basipterygoid, and quadrate rami. The pterygoid has no contact with the exoccipital. From the anterior part of the quadrate ramus to the posterior part of the palatine ramus, the lateral border of the pterygoid bears a strip of polygonal ornamentation. Almost the entire ventral surface of the palatine ramus presents a shagreen field except for its most posterolateral part, as previously mentioned. The palatine ramus forms a long, wide and thick sheet of bone. It is anterolaterally directed, with the anterior extremity contacting the ectopterygoid and the jugal. On the anterior tip of both rami a part of the ventral side is missing, creating an empty space. On the left ramus, a small patch of dot-shaped ornamentation anterior to the above-mentioned empty space indicates the anterior limit of the pterygoid. In anterior direction, the rami become slightly wider until their contact with the jugal. Then, their lateral borders become straight and anteriorly directed. The medial edge of each ramus forms the posterior half of the lateral border of the interpterygoid vacuity. It is slightly curved near the basipterygoid ramus and then become nearly straight anteriorly.

The basipterygoid ramus is very short and laterally wide compared to the other rami. On the medial side it has a complex, tight and dorsoventrally elongated suture with the parasphenoid bone, only visible from the ventral side. A deep and well-delimited subconical depression, the conical recess, is visible on the dorsal portion of this branch. This depression is laterally delimited by the anterior base of the lamina ascendens and posteriorly by the crista parapterygoidea of the parasphenoid. This is a similar condition to that found in Lyrocephaliscus euri [START_REF] Mazin | L'anatomie de Lyrocephaliscus euri (Wiman), trématosaure du Trias Inférieur du Spitsberg: Arrièrecrane, squelette axial et ceinture scapulaire[END_REF] and personal observations of T.A. on MNHN F SVT 520), both in terms of shape and location on the basipterygoid ramus.

On the ventral surface of each basipterygoid ramus, a pair of grooves is visible, with an anterior one reaching the interpterygoid fenestra and a posterior one reaching the posterior border of the basipterygoid ramus (Fig. 1,2). These grooves are nearly straight, with an antero-posterior orientation, and being between 3 and 8 mm wide. In addition, based on microCT data, the anterior one is connected with the posterior one by a canal inside the bone. Similar grooves are found in several species of temnospondyls, running through the whole length of the basipterygoid rami (e.g. Lyrocephaliscus euri, Mazin and Janvier 1983; Thoosuchus yakovlevi; personal observations of T.A. on specimen SMNS 58880; Trematolestes hagdorni, [START_REF] Schoch | A complete trematosaurid amphibian from the Middle Triassic of Germany[END_REF].

The quadrate ramus of the pterygoid forms a bony blade, about two times shorter than the palatine ramus. It shares a tight suture with the quadrate (but the left quadrate is missing) and it is dorsally in contact with the lamina ascendens. The quadrate ramus is slightly inclined medially and transversally oriented. The lateral edge of this ramus is concave and forms the posteromedial part of the subtemporal vacuity.

The lamina ascendens of the pterygoid is a slender, subvertical bony lamella, slightly thicker anteriorly. It medially delimits the columellar cavity in extending the lamina descendens of the squamosal, but these two laminae remain separated by a space, the palatoquadrate fissure. This structure lies on the quadrate and the basipterygoid rami with a transversal orientation except for its anterior tip, which points medially. At the level where the lamina ascendens contacts the quadrate ramus, a crest is present on the medial side, called 'crista obliqua' by Bystrow and Efremov (1940: fig. 5). It delimits a large, long and shallow canal on the medial side of the lamina ascendens.

The ventral surface of the pterygoid presents three kinds of ornamentation. In ventral view, the medial part of the basipterygoid ramus and the margin delimiting the interpterygoid vacuities are smoothed. This ornamentation pattern on the ventral surface of the pterygoid is also found in several trematosaurs (e.g. Thoosuchus yakovlevi, [START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF]Lyrocephaliscus euri, Mazin and Janvier 1983 and personal observations of T.A. on MNHN F SVT 520; Trematosaurus galae, Novikov 2010), presenting variation in the relative extension of the ornamentation.

The parasphenoid is divided in two parts: the cultriform process anteriorly and the basal plate posteriorly. The cultriform process is a long and narrow rod, laterally compressed, connecting the anterior part of the basal plate to the vomers.

It makes the separation between the two interpterygoid vacuities.

On the dorsal side of the cultriform process, two crests, the 'cristae laterosphenoidales' sensu [START_REF] Schoch | Comparative osteology of Mastodonsaurus giganteus (Jaeger, 1828) from the Middle Triassic (Lettenkeuper: Longobardian) of Germany (Baden-Württemberg, Bayern, Thüringen)[END_REF] are visible. They are more marked in the posterior part of the cultriform process. The ventral side of the cultriform process is convex in almost its entire length, with a strong lateral compression that gives a "knife-edge" shape to the cultriform process.

The basal plate of the parasphenoid is almost rectangular (length of 18.1 mm and width of 15.3 mm) with a pointed anterior side and a posterior projection (only visible in ventral view because it is covered by the exoccipital in dorsal view). Laterally, the basal plate shares a complex suture with the pterygoids, a typical stereospondyl character (e.g. [START_REF] Yates | The phylogeny of the 'higher' temnospondyls (Vertebrata: Choanata) and its implications for the monophyly and origins of the Stereospondyli[END_REF], and posteriorly with the exoccipital.

The anterior margin of the basal plate of the parasphenoid is slightly concave and forms the posterior limit of the interpterygoid vacuities and the beginning of the cultriform process.

On the dorsal surface of the basal plate of the parasphenoid, two crests are posterolaterally located, named 'cristae parapterygoideae' sensu [START_REF] Bystrow | Benthosuchus sushkini: A labyrinthodont from the Eotriassic of Sharzhenga River[END_REF] fig. 9A). They have a transverse orientation, like in Lyrocephaliscus euri (Mazin and Janvier 1983: fig. 7; personal observations of T.A. on MNHN F SVT 520).

Lateral to the ventral side of the posterior projection there is a pair of well-separated muscular pockets, as in Thoosuchus yakovlevi [START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF]: they show a curved anterior border, weakly marked, and open posteriorly.

Regarding the interpterygoid vacuities, they are placed in middle palatal position and occupy more than half of the palate. Each vacuity forms an elongated oval with a length of 63 mm (right one) and 64.1 mm (left one) and a width of 16.2 and 15.3 mm respectively. The anterior margin of the interpterygoid vacuities is more pointed than the posterior one.

The subtemporal vacuity is oval with an elongated anterior projection, more pointed than the posterior end of the vacuity. Its total length is 37.6 mm (right), and 34.6 mm (left) and maximum width of 19.3 mm (right) and 18.6 mm (left). The margins of this vacuity are formed by the pterygoid, which forms the major part of it, the jugal, the quadratojugal and the quadrate. The jugal, the quadratojugal and the quadrate are difficult to observe from this view because they are broken.

Dentition

The dentition is characterized by small teeth, uniform in morphology and size, and six fangs that are bigger than the other teeth. These fangs are usually aligned with rows of small teeth. Both of the fangs and the smaller teeth are labyrinthodont. There is one fang of about 5 mm in diameter between each anterior palatal vacuity and the choana. This fang, placed on the vomer, precedes a row of between 5 and 7 small teeth (Figs. 1,2) that surrounds the internal border of each choana. The exact number of teeth that forms the row is difficult to determinate due to poor preservation. Between the two vomers, closely placed on the upper part of the choana, there is a slight bone elevation with a V-shaped morphology. The presence of teeth in this area is doubtful but cannot be discarded. There are two additional fangs on the palatines. One palatine fang is lost. The other palatine fang, with a diameter of about 4 mm, is followed by a row of 4-6 small teeth. After this row of teeth, on the ectopterygoid, there is another badly-preserved fang. This fang is closely placed to a last row of small teeth that goes through the ectopterygoid. This row is longer than the previous ones but the exact number of teeth cannot be assessed.

Occipital region

The exoccipital is formed by the columna verticalis, bearing the articular condyle, and four processes: the processus submedullaris, the processus lamellosus, the processus paroticus and the processus subtympanicus (Figs. [START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]2,3). The exoccipital delimits two openings on the skull roof: the foramen magnum has a keyhole shape (14 mm × 9 mm at maximum size) as in Thoosuchus yakovlevi, and a postemporal fenestra with an isosceles subtriangular shape with the tip laterally oriented (6 mm in height × 11 mm in width for the right one and 6 mm × 10 mm for the left one). For this postemporal fenestra, the shape is the same as in Thoosuchus yakovlevi, although it is larger in Angusaurus. The size of the postemporal fenestra corresponds to 33% of the skull height and 26% of the skull length in Angusaurus, against 20 and 22% respectively in Thoosuchus yakovlevi (personal observations of T.A. on specimen SMNS 58880). Skull height is measured as the distance between the skull roof and the base of the parasphenoid at the level of the suture between the two postparietals, whereas skull length corresponds to the distance between the two tabular horns (since a part of the left cheek is missing in Angusaurus, the distance between the two cheeks was not used). Once again, Angusaurus and Thoosuchus yakovlevi present similar skull shape.

The columna verticalis contacts a posterior column-like projection of the postparietal dorsally. Beginning from the ventral side, the suture between the exoccipital and the postparietal is located at three quarters of the occipital height (Fig. 1e,f,3c).

The processus submedullaris is located medially on the columna verticalis, at one-third of the height of the exoccipital (Fig. 3a). The dorsal surface of the processus submedullaris is concave with a truncated oval shape but the anterior border is straight, similar to Benthosuchus sushkini (Bystrow and Efremov 1940: fig. 11).

The processus lamellosus is located above the processus submedullaris and medial to the columna verticalis, at around mid-height of the foramen magnum. It is relatively poorly developed compared to Benthosuchus sushkini (Bystrow and Efremov 1940: fig. 11). The dorsal surface of this process is concave with an oval shape.

The processus paroticus is projects dorsolaterally from the columna verticalis and forms half of the ventral part of the paroccipital process with the descending flange of the tabular. Posterolaterally, a wide foramen is present as in Benthosuchus sushkini (Bystrow and Efremov 1940: fig. 11) or Lyrocephaliscus euri (Mazin and Janvier 1983: fig. 4.)

The processus subtympanicus is an anterolaterally-projected sheet of bone. It is rectangular, but the left one has a rounded anterior extremity that makes it longer than the right one.

The occipital condyles lie in posteromedial position in relation to the columna verticalis. They are as high as wide and well separated from each other. The posterior and medial surfaces are rough.

On the anterior side of the exoccipital, between the base of the processus lamellosus and paroticus, there is a weak rounded depression. Due to this emplacement and by comparison with Stanocephalosaurus amenasensis [START_REF] Arbez | Exceptional endocranium and middle ear of Stanocephalosaurus (Temnospondyli: Capitosauria) from the Triassic of Algeria revealed by micro-CT scan, with new functional interpretations of the hearing system[END_REF], it could represent the opisthotic bone.

Preservation of internal structures

Unfortunately, the stapes, otic capsule, epipterygoid, sphenethmoid and basisphenoid are not preserved in SMNS 81782. However, this lack of preservation is not surprising, since the inner structures of stereospondyls are generally poorly ossified, likely cartilaginous, and thus rarely preserved in the fossil record. In trematosaurs, the sphenethmoid is only present in Aphaneramma sp. [START_REF] Säve-Söderbergh | On the morphology of Triassic stegocephalians from Spitsbergen, and the interpretation of the endocranium in the Labyrinthodontia[END_REF] and Benthosuchus sushkini [START_REF] Bystrow | Benthosuchus sushkini: A labyrinthodont from the Eotriassic of Sharzhenga River[END_REF], and only Aphaneramma sp. is known to have a preserved basisphenoid (Säve-Söderbergh 1936).

Moreover, in species where the stapes, the otic capsule and the epipterygoid are preserved, these bones possess few and slight contacts with surrounding bones. For example, in S. amenasensis the epipterygoid is only linked to the pterygoid and likely by a cartilaginous contact [START_REF] Arbez | Exceptional endocranium and middle ear of Stanocephalosaurus (Temnospondyli: Capitosauria) from the Triassic of Algeria revealed by micro-CT scan, with new functional interpretations of the hearing system[END_REF]). As explained above, because of their weak ossification these elements are easily lost during fossilisation.

Phylogenetic analyses

The phylogenetic position of the genus including all species of Angusaurus was assessed via cladistic analyses. We used the data matrix of [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF], which is based on [START_REF] Steyer | The first articulated trematosaur "amphibian" from the lower Triassic of Madagascar: Implications for the phylogeny of the group[END_REF], but includes some coding changes due to observation of new specimens as well as additional taxa. Herein, we added two more taxa (Syrtosuchus samarensis, Prothoosuchus blomi) based on the literature [START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF][START_REF] Novikov | New temnospondyl amphibians from the Lower Triassic of Obshchy Syrt (Eastern Europe)[END_REF]. We tested the genus Angusaurus in two different analyses. In one of the analyses, we coded the three species A. dentatus, A. succedaneus and A. tsylmensis separately, while in the second analysis we coded all three Angusaurus species as a group (labeled as Angusaurus spp.). All the terminal taxa were considered at the specific level (except for the Angusaurus cluster). Onchiodon frossardi and Archegosaurus decheni were considered as outgroups. In total, 25 terminal taxa and 69 characters were analyzed (see supplementary material for the data matrix) using the software PAUP 4.0 beta for PC [START_REF] Swofford | PAUP: Phylogenetic Analysis Using Parsimony[END_REF]. All characters were considered equally weighted and unordered. Eight characters from the data matrix were found to be uninformative, and excluded from the analyses. These excluded characters are (same as in the previous published analysis, [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF]): 16 (frontal contact orbit), 17 (interfrontal), 28 (vomer contact maxilla), 46 (paraquadrate foramen), 56 (stapes morphology), 62 (parasymphysial tooth row), 63 (denticles coronoids) and 67 (crista articularis). We performed a heuristic search of 10,000 replicates with simple addition sequence and only one tree retained at each step during stepwise addition. Tree-Bisection-Reconnection (TBR) branch swapping was performed and zero-length branches were collapsed to yield polytomies.

The analysis treating the three Angusaurus species (A. dentatus, A. succedaneus, A. tsylmensis) as separate species, recovered five most parsimonious trees (MPT) of 202 steps (CI 0.396, RI 0.4831, RC 0.1913) (Fig. 4). The consistency index is low and reveals a high amount of homoplasy in trematosaur evolution. A bootstrap analysis with 1000 replicates also reveals relatively low support, with only two clades with support over 50% (Stereospondyli: 61% and Aphaneramma rostratum + Aphaneramma kokeni: 51%). The monophyletic Lonchorhynchinae (node 12) is supported by four synapomorphies: orbits facing laterally [5(1)], knifeedged cultriform process [38(2)], absence of the crista muscularis of the parasphenoid [42(0)], and carotid canal visible on the ventral surface of the parasphenoid [43 [START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]].

In this analysis, Lonchorhynchinae is the sister-taxon of Trematosaurinae (node 15), as found by [START_REF] Steyer | The first articulated trematosaur "amphibian" from the lower Triassic of Madagascar: Implications for the phylogeny of the group[END_REF] and [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF]. The Trematosaurinae clade is composed by the same taxa as in previously recovered trees [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF]), but with some taxa in different positions (see below). Trematosaurinae is supported by twelve synapomorphies: orbits facing laterally [5(1)], nostril in lateral position [9(1)], pineal large relative to the size of the skull roof [11(1)], frontal extended behind orbit [15(1)], premaxilla/nasal suture straight [24(1)], ventral opening of the orbit in the anterior half of the interpterygoid vacuity [27(2)], presence of paired anteropalatal vacuities [31(2)], anteropalatal vacuities between the premaxilla/vomerine suture [32( 1 The taxon Syrtosuchus samarensis [START_REF] Novikov | New temnospondyl amphibians from the Lower Triassic of Obshchy Syrt (Eastern Europe)[END_REF]) is here found as the most basal taxon of the analyzed taxa. [START_REF] Novikov | New temnospondyl amphibians from the Lower Triassic of Obshchy Syrt (Eastern Europe)[END_REF] erected this taxon and considered it "as the most primitive benthosuchid taxon assigned to a separate benthosuchid subfamily, Syrtosuchinae(…)" (Novikov 2016: p. 307). In the present analysis this taxon represents a stem trematosauroid, and potentially a stem capitosauroid. Future analyses focusing on basal trematosauroids and related taxa should clarify the position of this basal taxon and assess its phylogenetic position.

Lonchorhynchinae is composed by the same taxa (and equal positions) found by [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF], while Trematosaurinae is composed by the same taxa as in [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF], with the addition of Prothoosuchus blomi (not included in previous analyses). Trematolestes hagdorni is found as the most basal trematosaurine, representing the sister taxon of Thoosuchus yakovlevi. In turn, this latter taxon is the sister taxon of the clade formed by all three Angusaurus species analyzed as well as Prothoosuchus blomi, resulting in a polytomy between these four taxa (node 7, Fig. 4). It should be mentioned that a heuristic analysis (under the same variables) but not including Angusaurus succedaneus resulted in only one Most Parsimonious Tree (MPT) with the same topology as above but with Angusaurus dentatus recovered as the sister taxon of the clade composed by Angusaurus tsylmensis and Prothoosuchus blomi (Fig. 4). In a similar way, the analysis grouping all three Angusaurus species (labeled as Angusaurus spp.) resulted in only one MPT with the same general topology but recovering a clade formed by Angusaurus spp. and Prothoosuchus blomi.

Regarding the rest of trematosaurines, this latter analysis recovered a clade (node 6) composed by Trematosaurus brauni and Trematosuchus sobeyi, not recovered in previous analyses by [START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF] but present in analyses made by [START_REF] Steyer | The first articulated trematosaur "amphibian" from the lower Triassic of Madagascar: Implications for the phylogeny of the group[END_REF]. Finally, Inflectosaurus and Microposaurus appear as derived trematosaurines [START_REF] Damiani | Cranial anatomy and relationships of Microposaurus casei, a temnospondyl from the Middle Triassic of South Africa[END_REF][START_REF] Novikov | New data on trematosauroid labyrinthodonts of Eastern Europe: 1. Genus Inflectosaurus Shishkin[END_REF][START_REF] Warren | The South African stereospondyl Microposaurus from the Middle Triassic of the Sydney Basin, Australia[END_REF][START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF]) and closely related to Platystega, Lyrocephaliscus and Tertrema.

Discussion

Taxonomic and ontogenetic assessment of SMNS 81782

Trematosauroids are characterized by the formation of the anterior portion of a long and narrow cultriform process from the posterior projections of the vomer [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF], a feature also found in thoosuchids and in the studied specimen. In the specimen described neither the frontal nor the lacrimal bones have any contact with the orbits, a derived feature also present in Thoosuchus and other trematosauroids [START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF]. In capitosauroids the lacrimal does not contact the orbits but the frontal maintains its orbit contact, causing a wide separation between the prefrontal and the postorbital, like in Edingerella [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF]Maganuco et al. 2009).

In the trematosaurines Thoosuchus and Angusaurus, as well as SMNS 81782, the supraorbital sulcus runs anteriorly through the nasal, between the nares, and then enters the lacrimal, whereas in most capitosauroids this sulcus runs along almost the entire nasal and enters the lacrimal posteriorly (Fig. 1; [START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF]Maganuco et al. 2009;[START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF]. The presence of the occipital canal is typical of many trematosaurs like Thoosuchus and Angusaurus, while its absence is characteristic of capitosauroids, like Edingerella (Maganuco et al. 2009).

Small trematosaurids exclusively known from the Lower Triassic of the Eastern European Platform are usually referred to Thoosuchidae. This group is characterized by a narrow skull with concave lateral margins and a longer snout than in other trematosauroids, and by the lateral margin of the skull being straight or slightly convex like in Benthosuchus [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF]Novikov 2012a). All these features are also true for the specimen studied herein.

The flexura lacrimalis is represented by a Z-shaped flexura on the lacrimal of the infraorbital sulcus. This flexura is present in thoosuchids, such as Thoosuchus and Angusaurus, as well as in the specimen described, being softer and less sharp in the latter than in other trematosauroids like Benthosuchus [START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF][START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF][START_REF] Damiani | The Triassic amphibian Thoosuchus yakovlevi and the relationships of the Trematosauroidea (Temnospondyli: Stereospondyli)[END_REF]Novikov 2012a).

The jugals of thoosuchids are thinner in comparison with other trematosauroids. These features can be also shared with primitive trematosauroids. In adult benthosuchids, the jugal can have the width of its orbits [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF]. The palatine of thoosuchids, as well as the specimen analyzed here, has no posterior process or contact with its pterygoid, whereas the palatines of Benthosuchus connects with the pterygoid (Novikov 2012a;[START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF].

Taking into account all these features and the fact that the specimen SMNS 81782 cannot be referred to Benthosuchidae-because members of this family have a bilobed anterior palatal vacuity-, nor to Plastystegididae-because its members have a very broad postorbital skull-this specimen is assigned with confidence to Thoosuchidae (sensu [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF]. Regarding Thoosuchus, the specimen cannot be referred to this genus based on orbit size (bigger in Thoosuchus) as well as the elongation of the postorbital and postfrontal in the studied specimen relative to Thoosuchus. The genus Angusaurus (Figs. [START_REF]Orbit facing laterally (1) or dorsally[END_REF][START_REF]) or oval (1) orbit[END_REF][START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF], as previously mentioned, is characterized by posteriorly elongated postorbitals and postfrontals [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF], as in the studied specimen. The main part of the body of the jugal of the studied specimen is situated in a posterior position, behind the orbits, [START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF]. The body of the parasphenoid of Angusaurus is slightly longer [START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF]) than in the studied specimen (Figs. [START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]2,3,[START_REF]) or oval (1) orbit[END_REF]. As in SMNS 81782, the pineal foramen of the genus Angusaurus is situated at half the distance between the orbits and the occipital, in a middle position on the parietals [START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF]. All these characters allow us to assign the studied specimen to the genus Angusaurus.

To date, four Angusaurus species have been erected: A. dentatus, A. succedaneus,A. weidenbaumi and A. tsylmensis (Figs. 5,[START_REF]) or oval (1) orbit[END_REF][START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF]. Unfortunately the diagnostic characters for each one of these species are tenuous. In addition, some of the specimens used in published descriptions are differently preserved in the homologous regions, which precludes in some cases anatomical comparison. A. dentatus (type species of the genus) and A. tsylmensis are erected based on almost complete skulls (but missing some parts of the pterygoids and parasphenoid in A. dentatus, as well as any mandible remain known for this taxon) (Figs. [START_REF]Orbit facing laterally (1) or dorsally[END_REF][START_REF]) or oval (1) orbit[END_REF][START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF] whereas A. weidenbaumi is only known by a snout (not presenting the typical flexura lacrimalis sulcus found in the rest of Angusaurus taxa) and A. succedaneus by a partial skull with well-preserved palate and mandible but missing most of the skull roof and occiput, but associated with a well preserved mandible (Figs. [START_REF]Orbit facing laterally (1) or dorsally[END_REF][START_REF]) or oval (1) orbit[END_REF][START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF].

Hence, the key anatomical characters in the analyzed specimen are: (1) (character 36; see supplementary material) posterior branch of the pterygoid, narrow and elongate (A. dentatus) or short and wide (A. tsylmensis and SMNS 81782); (2) presence of flexura lacrimalis (A. dentatus, A. succedaneus, A. tsylmensis and SMNS 81782) or absence (A. weidenbaumi); (3) the anterior and lateral border of the basal plate of the parashphenoid are almost straight (A. dentatus) or concave (A. succedaneus, A. tsylmensis and SMNS 81782); (4) ventral surface of the parasphenoid plate smoothed (A. dentatus and SMNS 81782) or dot-shaped (A. succedaneus, A. tsylmensis); (5) postemporal fenestra high (the fenestra itself, not its position) (A. dentatus and SMNS 81782) or low (A. tsylmensis).

Nonetheless, the last two characters should be taken with caution: the ornamentation is known to vary with ontogeny [START_REF] Steyer | Ontogeny and phylogeny in temnospondyls: A new method of analysis[END_REF], and the height difference of the fenestra is not really pronounced and cannot be excluded for preservation reasons (compression).

Overall, all these characters suggest affinities of the studied specimen with A. tsylmensis or A. succedaneus, and prevent the assignment of the former to A. dentatus and A. weidenbaumi. However, an attribution to A. succedaneus is not possible to conclude due to the poor state of preservation of some diagnostic characters and anatomical areas (e.g. quadrate ramus) in the holotype (and unique published specimen) of A. succedaneus, and it cannot be discarded that some shared characters with this taxon may be due to ontogeny. Thus, based on the shared characters discussed, SMNS 81782 is referred to as A.cf. tsylmensis. The lack of information on the stratigraphy of the studied [START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF][START_REF] Novikov | New Early Triassic labyrinthodonts of the Middle ForeUrals[END_REF] and personal observations (J.F.). Note that lines for cranial sutures in A. succedaneus are tentative specimen, as well as potential interspecific, intraspecific, ontogenetic or preservation variability preclude a confident assignment to A. tsylmensis.

As previously mentioned, the approximate length of the adult skull for Angusaurus is about 200 mm (Getmanov 1989), while the studied skull has a length of 112.3 mm, indicating a possible juvenile condition. The radial sculpture of the dermal bones is typically found in growing bones and it can thus be a larval or juvenile character, although this ornamentation can also be found in some adult stereospondyls. Polygonal sculpture comes from the radial sculpture pattern [START_REF] Witzmann | Sculpture and vascularization of dermal bones, and the implications for the physiology of basal tetrapods[END_REF], and SMNS 81782 has mainly polygonal ornamentation in the ossification centre of the skull roof. Although we interpret SMNS 81782 as a sub-adult, we cannot reject that this specimen could be an adult considering that the radial pattern can also be present in adults. Sometimes the degree of ossification and the sutural fusion is indicative of a developmental age but in temnospondyls it might not be a significant indicator [START_REF] Steyer | Ontogeny and phylogeny in temnospondyls: A new method of analysis[END_REF].

Angusaurus: taxonomic and phylogenetic considerations

As previously mentioned, four Angusaurus species have been erected to date (A. dentatus, A. succedaneus, A. weidenbaumi and A. tsylmensis) (Figs. [START_REF]Orbit facing laterally (1) or dorsally[END_REF][START_REF]) or oval (1) orbit[END_REF][START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF]. This genus is apparently uniform, with few diagnostic characters allowing differentiation between species. This fact raises the question of the validity of these species. When three species (A. dentatus, A. succedaneus, and A. tsylmensis) are included separately in the phylogenetic analysis, it results in a polytomy (Fig. 4). In particular, two of these species (A. dentatus and A. succedaneus) are virtually equal under the character matrix (depending on the missing data), and it should be considered that both species were recovered from the same subhorizon.

Previously, [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF] also questioned the validity of A. succedaneus, suggesting it as potential junior synonym of A. dentatus. Nonetheless, these authors maintained the validity of these species.

The main reason why we do the same here is because of the difficulty to directly compare A. dentatus, A. succedaneus, and A. weidenbaumi (see above). Although these three taxa are recovered from the same subhorizon (see below), there is a lack of homologous regions preserved in these three species. However, we cannot discard the possibility that future findings may potentially reveal that only one or two of these species are valid, thus raising the possibility of evaluating the intraspecific variability of the genus Angusaurus.

On the other hand, A. tsylmensis is considered more confidently as a valid species because it was recovered from a different (younger) subhorizon (Ust-Mylian) and presents the morphological characters previously discussed (e.g. short and wide posterior branch of the pterygoid; the more anterior position of the quadrate condyle, etc.) (Figs. 5,[START_REF]) or oval (1) orbit[END_REF][START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF].

Biostratigraphy and paleobiogeography of basal trematosauroids, trematosaurids and trematosaurines

Most trematosaurines have been recovered from the Russian European platform, with the exception of some taxa (e.g. Trematolestes hagdorni, Trematosaurus brauni, Trematosuchus sobeyi etc.) from Central Europe, Spitsbergen, South Africa, Madagascar, North America and Australia.

Regarding the communities of Eastern Europe, three different main regions (or realms) can be recognized: (a) the central and northern regions of the East European Platform (Moscow-Mezen, and presumably Pechora, Synclines), and the Timan-North Ural Region; (b) the southern CisUrals, including the Obshchii Syrt Plateau; and (c) the southern regions of the East European Platform (the slope of the Voronezh Anticline) [START_REF] Shishkin | Differentiation of tetrapod communities and some aspects of biotic events in the Early Triassic of Eastern Europe[END_REF].

From a chronological point of view, these main regions span much of the Lower and Middle Triassic (from the Induan to the Ladinian), with a succession up to 3.5 km and subdivided in different superhorizons (or gorizonts, here used as synonyms but see [START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF] for discussion): Vetlugian superhorizon (Induan-Olenekian), Yarenskian horizon (Olenekian), Donguz horizon (Anisian) and Bukobay horizon (Ladinian). Up to seven faunal assemblages (or svitas, here used as synonyms but see [START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF] for discussion) have been proposed for these superhorizons and horizons: Kopanskaya (Induan), Staritskaya, Kzylsaiskaya, Gostevskaya, and Petropavlovskaya (all Olenekian), Donguz (Anisian), and Bukobay (Ladinian) (see [START_REF] Ochev | Hierarchy of Lower Triassic stratigraphic units in the eastern part of European Russia[END_REF]Tverdokhlebov et al. 2002: fig. 2;[START_REF] Shishkin | Differentiation of tetrapod communities and some aspects of biotic events in the Early Triassic of Eastern Europe[END_REF] for comprehensive reviews, correlation between horizon and assemblages, as well as additional references). Herein, we focus on basal trematosauroids, trematosaurids and trematosaurines recovered from the Vetlugian superhorizon and associated with the Benthosuchus-Wetlugasaurus tetrapod complexes. This superhorizon spans the following subhorizons (or gorizonts), from base to top: Vokhmian, Rybinskian, Sludkinskian and Ust-Mylian [START_REF] Ochev | Hierarchy of Lower Triassic stratigraphic units in the eastern part of European Russia[END_REF] fig. 1 for a discussion). Vetlugian superhorizon is dated as Olenekian with the exception of the Vokhmian subhorizon (Induan). This latter subhorizon is of particular interest for trematosauroids, since different taxa (as part of the Kopanskaya faunal assemblage) have been recovered from this subhorizon and particularly from the main region of the southern CisUrals and the Obshchii Syrt Plateau. Remarkably, Syrtosuchus and primitive Benthosuchus taxa (e.g. B. gusevae) have also been recovered, as well as other temnospondyls such as Tupilakosaurus and Wetlugasaurus samariensis [START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF]Novikov 2012a[START_REF] Novikov | New temnospondyl amphibians from the Lower Triassic of Obshchy Syrt (Eastern Europe)[END_REF].

Nonetheless, a recent proposed new horizon, Zaplavnenskian Horizon, has been proposed. Its faunal assemblage is still controversial but suggests a late developmental stage of the Tupilakosaurus fauna (see Shishkin and Novikov 2017 and references therein).

The Rybinskian Gorizont (Staristskaya faunal assemblage) is mainly composed of fluvial sandstones, with the usual features of arid zones [START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF]. Benthosuchus sushkini and the genus Thoosuchus are the most characteristic taxa for this gorizont, forming a Benthosuchus-Thoosuchus group, typical of this subhorizon but also including other temnospondyl members as Wetlugasaurus, Trematotegmen otschevi and Qantas samarensis [START_REF] Shishkin | Tetrapod biostratigraphy of the Triassic of Eastern Europe[END_REF][START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF][START_REF] Novikov | New data on trematosauroid labyrinthodonts of Eastern Europe: 3. Qantas samarensis gen. et sp[END_REF]). The Sludkinskian subhorizon (Kzylsaiskaya faunal assemblage) is very similar to the previous one, but in the uppermost part of the section the fluvial deposits are sometimes replaced by deltaic sediments. The fauna is similar to the Rybinskian subhorizon, particularly on the lowermost part of the section. In this faunal assemblage Wetlugasaurus angustifrons is the most common tetrapod, but it also contains other temnospondyls like Wetlugasaurus kzilsajensis, Benthosuchus bashkiricus, Thoosuchus tuberculatus, Angusaurus dentatus, Angusaurus succedaneus, Angusaurus weidenbaumi, Prothoosuchus blomi, and Prothoosuchus samariensis, and the first appearance of the capitosaur genus Parotosuchus [START_REF] Shishkin | Tetrapod biostratigraphy of the Triassic of Eastern Europe[END_REF][START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF].

Finally, the Ust-Mylian subhorizon (Gostevskaya faunal assemblage and equivalent to the assemblage III of Novikov 1994) constitutes a complex of lacustrine-deltaic accumulations where the most characteristic temnospondyl fauna is composed of Wetlugasaurus malachovi as well as Angusaurus tsylmensis and Vyborososaurus mirus. It also represents the first appearance of the genus Trematosaurus as well as the capitosaur Vladlenosaurus alexeyevi and indeterminate plagisaurids [START_REF] Ochev | Hierarchy of Lower Triassic stratigraphic units in the eastern part of European Russia[END_REF][START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF][START_REF] Shishkin | Differentiation of tetrapod communities and some aspects of biotic events in the Early Triassic of Eastern Europe[END_REF]). It should be remarked that this faunal assemblage is the most poorly known due to difficulties recognizing this fauna outside of the northeastern areas of the Russian platform.

Later, the temnospondyl fauna was replaced by the Parotosuchus group, which dominated the Yarenskian subhorizon [START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF].

Taking into account these previous considerations, the Early Olenekian time is of particular interest to understand the origin and diversification of basal trematosaurines. This time is characterized by a eustatic maximum [START_REF] Lozovsky | The Early Triassic Stage in the development of Western Laurasia[END_REF], which resulted in the expansion of lakes in Eastern Europe [START_REF] Shishkin | Differentiation of tetrapod communities and some aspects of biotic events in the Early Triassic of Eastern Europe[END_REF]) and represented by two subhorizons corresponding to the Early Olenekian: the Sludkinskian (early Early Olenekian) and the Ust-Mylian (late Early Olenekian) [START_REF] Ochev | Hierarchy of Lower Triassic stratigraphic units in the eastern part of European Russia[END_REF][START_REF] Tverdokhlebov | Tetrapod localities from the Triassic of the SE of European Russia[END_REF]. The chronological origin of trematosauroids, trematosaurids and trematosaurines is debated, since the earliest fossil evidence for these groups ranges from (?Late) Induan to Early Olenekian, known from different continents. On the one hand, basal trematosauroids such as Syrthosuchus sp. and Benthosuchus gusevae are recorded from the (?Late) Induan (Vokhmian subhorizon) from the Southern Cis-Ural area-Obshchii Syrt Plateau (Novikov 2012a[START_REF] Novikov | New temnospondyl amphibians from the Lower Triassic of Obshchy Syrt (Eastern Europe)[END_REF]. Nonetheless, the poorly known genus Tirraturhinus from the Arcadia Formation (central Queensland, Australia) has been dated as (Early) Induan and proposed as a trematosaurine [START_REF] Nield | A short-snouted trematosauroid (Tetrapoda, Temnospondyli) from the Early Triassic of Australia: The oldest known trematosaurine[END_REF][START_REF] Warren | The South African stereospondyl Lydekkerina huxleyi (Tetrapoda, Temnospondyli) from the Lower Triassic of Australia[END_REF]; but see discussion about alternative dating and taxonomic position in [START_REF] Shishkin | Differentiation of tetrapod communities and some aspects of biotic events in the Early Triassic of Eastern Europe[END_REF] and [START_REF] Novikov | New data on trematosauroid labyrinthodonts of Eastern Europe: 3. Qantas samarensis gen. et sp[END_REF].

On the other hand, recent phylogenetic analyses place the genus Trematolestes as the most basal trematosaurine (Fortuny et al. 2018, present study), but also resembling in some characters lonchorhynchines [START_REF] Schoch | A complete trematosaurid amphibian from the Middle Triassic of Germany[END_REF].This wellknown taxon is recorded from the Middle Triassic (Lower Keuper facies, Upper Ladinian) of Germany and is closely related to the Early Triassic Malagasy genus Tertremoides [START_REF] Schoch | A complete trematosaurid amphibian from the Middle Triassic of Germany[END_REF]. If this is correct, Tertremoides, recovered from the NW of Madagascar (Ankitokazo basin, Sakamena Group, Ambilobe region), has to be considered a basal trematosaurine. This genus is found in similar levels among other trematosaurids such as Wantzosaurus and Aphaneramma [START_REF] Steyer | The first articulated trematosaur "amphibian" from the lower Triassic of Madagascar: Implications for the phylogeny of the group[END_REF][START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF]). The Ankitokazo basin was originally dated as Induan (Scythian A1), but later redated as Olenekian based on conchostracans [START_REF] Yanbin | Studies on Permo-Trias of Madagascar. 4. Early Triassic conchostracans from Madagascar[END_REF].

Overall, the first trematosauroid datum implies an open paleobiogeographical scenario where basal trematosauroids may have originated during the Induan at the Cis-Uralian-Obshchii Syrt Plateau region of the Eastern Europe platform [START_REF] Novikov | New temnospondyl amphibians from the Lower Triassic of Obshchy Syrt (Eastern Europe)[END_REF]), but without excluding the possibility of an Australian [START_REF] Nield | A short-snouted trematosauroid (Tetrapoda, Temnospondyli) from the Early Triassic of Australia: The oldest known trematosaurine[END_REF] or East Greenland (see below) trematosauroid origin and/or even suggesting paleobiogeographical link between East Greenland, Cis-Uralian-Obshchii Syrt Plateau (Russia) and Queensland (Australia). Otherwise, the trematosauroid split between short-snouted forms (e.g. trematosaurines) and long snouted ones (lonchorhynchines), is particularly unclear based on the presence of the most basal members of these groups during the Late Induan-(?Early) Olenekian in different continents: East Greenland (Stoschiosaurus, Late Induan, [START_REF] Nielsen | The Permian and Eotriassic vertebrate-bearing beds at Godthaab Gulf (East Greenland)[END_REF][START_REF] Zaton | Boreal earliest Triassic biotas elucidate globally depauperate hard substrate communities after the end-Permian mass extinction[END_REF] [START_REF] Steyer | The first articulated trematosaur "amphibian" from the lower Triassic of Madagascar: Implications for the phylogeny of the group[END_REF]. Taking into account these data, we come up with, at least, two hypotheses: (A) the first appearance of basal trematosauroids and trematosaurids is older than current fossil record evidence suggests; (B) supposed basal lonchorhynchines and trematosaurines (as Stoschiosaurus or Tertremoides) may be basal trematosauroids instead of lonchorhynchines or trematosaurines, and consequently a taxonomical re-assignment and definition of these groups is required (provided that new material is found).

Lastly, the trematosaurine radiation that started in Eastern Europe with the appearance of Thoosuchus and subsequent related genera such as Angusaurus, Prothoosuchus and Vyborosaurus during the Early Olenekian potentially expanded to Central European basins (with paleobiogeographical connection between the southern regions of the East European Platform, [START_REF] Shishkin | Differentiation of tetrapod communities and some aspects of biotic events in the Early Triassic of Eastern Europe[END_REF] represented by the presence of the genus Trematosaurus in both areas [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF][START_REF] Novikov | New data on Trematosauroid Labyrinthodonts of Eastern Europe: 2. Trematosaurus galae sp. nov.: Cranial Morphology[END_REF]). The Trematosaurus radiation during the Late Olenekian also derived in the presence of advanced trematosaurines such as Inflectosaurus, Platystega, Tertrema and Lyrocephaliscus in the Eastern European platform and Spitsbergen (Svalbard), and also in southern latitudes such as South Africa (Karoo basin) during the Late Olenekian-Early Anisian, represented by Trematosuchus and Microposaurus [START_REF] Schoch | Stereospondyli. Handbuch der Paläoherpetologie, Teil 3B[END_REF].

Conclusions

The genus Angusaurus, present from the early Early Olenkian to late Early Olenekian in Russia, is a trematosaurine characterized by skulls about 200 mm long (adult size), small orbits (compared to the close genus Thoosuchus), elongated postorbitals and postfrontals, with a very reduced interchoanal tooth row and a ventral opening of the orbits in the middle part of the skull, presenting a mandible with an elongate postglenoid process (PGA) and well developed suprangular. The new specimen described herein sheds some light on the ontogeny of this genus, with implications for other trematosaurines. The genus Angusaurus includes four species with few key anatomical characters. While the type species, Angusaurus dentatus, and Angusaurus tsylmensis are well supported, the validity of Angusaurus succedaneus and Angusaurus weidenbaumi is controversial and new findings are required to fully evaluate their validity and better understand the intraspecific variability of this genus. Deciphering the evolution of trematosaurines during the Early Olenekian is key to understanding the origin and diversification of basal trematosaurines with implications to clarify the different paleobiogeographical scenarios that remain still open.

Fig. 1

 1 Fig. 1 Angusaurus cf. tsylmensis. Early Olenekian (Lower Triassic) of Russia (SMNS 81782) in dorsal (a, b), ventral (c, d), occipital (e, f) and lateral view (g)

Fig. 2

 2 Fig. 2 3D model of Angusaurus cf. tsylmensis (SMNS 81782). Matrix removed in dorsal (a), ventral (b), lateral (c) and occipital (d) views

Fig. 3

 3 Fig. 3 3D model of Angusaurus cf. tsylmensis (SMNS 81782). Neurocranium and inner parts (a), oblique view of the occiput (b) and interpretative drawing of the oblique view (c)

Fig. 4

 4 Fig. 4 Phylogenetic position of Angusaurus within the Trematosauroidea clade. Strict consensus tree from the five Most Parsimonius Trees (MPT) recovered under a heuristic analysis (see text) (a).

  )], knife-edged cultriform process of the parasphenoid[38(2)], absence of the crista muscularis of the parasphenoid[42(0)], absence of the crista falciformis of the squamosal, in occipital view[53(0)], and Meckelian foramen elongate [65[START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]]. The clade joining Trematosaurinae and Lonchorhynchinae is the Trematosauridae (clade 16). This clade is supported by five synapomorphies: nostril ovoid [8(1)], otic notch deep and open[20(1)], anteropalatal vacuities between the premaxilla/vomerine suture.[32(1)], occipital condyles widely separated from each other [50(0)], and welldeveloped crista medialis[66(1)]. As previously recovered by[START_REF] Fortuny | A new extreme longirostrine temnospondyl from the Triassic of Madagascar: Phylogenetic and palaeobiogeographical implications for trematosaurids[END_REF], Trematosauridae and Benthosuchus sushkini form the clade Trematosauroidea (node 17), which is supported by ten synapomorphies: small orbit [4(1)], quadrate condyle anterior to the occipital condyle[22(1)], interpterygoid vacuity not widened at all[33(2)], anterior branch of the pterygoid entirely wide [35[START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]], ventrally directed tabular in occipital view[48(1)], foramen magnum ventrally constricted[51(1)], presence of the crista falciformis of the squamosal in occipital view[53(1)], deep postemporal fenestra [54[START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]], posterior end of the mandible situated behind the quadrate condyle [57[START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]], and extended dentary symphysis [60[START_REF]Skull roof elongate (midline length > maximum width; 1) or not[END_REF]].

Fig. 5

 5 Fig. 5 The genus Angusaurus from the Early Olenekian (Lower Triassic) of Russia. Dorsal view of: A. dentatus (a), A. succedaneus (b), A. tsylmensis (c), A. weidenbaumi (d). Redrawn after Getmanov

Fig. 6

 6 Fig. 6 The genus Angusaurus from the Early Olenekian (Lower Triassic) of Russia. Ventral view of: A. dentatus (a), A. succedaneus (b), A. tsylmensis (c) and A. weidenbaumi (d).Redrawn after[START_REF] Getmanov | Triassic amphibians of the East European platform (family Benthosuchidae Efremov)[END_REF][START_REF] Novikov | New Early Triassic labyrinthodonts of the Middle ForeUrals[END_REF] and personal observations (J.F.)

Fig. 7

 7 Fig.[START_REF]Slightly concave (0) or semicircular (1) posterior margin of the skull roof. 8. Nostril rounded (0), ovoid[END_REF] The genus Angusaurus from the Early Olenekian (Lower Triassic) of Russia. Occipital view of: A. dentatus (a), A. succedaneus (b), A. tsylmensis (c) and mandible in labial (d) and lingual (e) side of A. succedaneus. Redrawn afterGetmanov 1989, Novikov 1990 and personal observations (J.F.)

  Definition All taxa sharing a more recent common ancestor with Angusaurus dentatus than with Thoosuchus yakovlevi. Emended diagnosis Skull up to 200 mm long. Oval orbits, laterally emplaced. Elongated postorbitals and postfrontals. Interchoanal tooth row strongly reduced. Ventral opening of the orbit in the mid part. Postglenoid process of the mandible elongated. Medial process of supraangular well developed.

		Type horizon Vetlugian Horizon with two subhorizons:
		Sludkinsk Subhorizon, early Early Olenekian (Early Tri-
		assic) for Angusaurus dentatus, Angusaurus succedaneus,
		Angusaurus weidenbaumi.Ustmylskian Subhorizon, late
		Early Olenekian (Early Triassic) for Angusaurus tsylmensis.
		Locus typicus Left shore of Samara river at Dolgiy Yar,
		Buzulukskij Rajon, Orenburgskaya Oblast (A. dentatus);
		Chapayevsk, Alexeyevka District, Kuybyshevskaja Oblast
	Temnospondyli Zittel 1887-1890	(A. succedaneus); Pleas near Kineshma, Furmanovskij
	Stereospondyli Zittel 1887-1890	Rajon, Ivanovskaya Oblast (A. weidenbaumi); Mala local-
	Trematosauria Romer, 1947 sensu Yates and Warren 2000	ity, Tsyl'ma River region, Komi Republic (A. tsylmensis).
	Trematosauroidea Säve-Söderbergh 1935 sensu Schoch	All of them eastern European Russia.
	2013
	Trematosauridae Watson 1919
	Trematosaurinae Watson 1919, sensu Fortuny et al. 2018

Angusaurus Getmanov 1989 Angusaurus cf. tsylmensis Getmanov 1989 Type species.-Angusaurus dentatus Valid species.-Angusaurus dentatus, Angusaurus weidenbaumi, Angusaurus succedaneus, Angusaurus tsylmensis.

  ; Eastern Europe (Thoosuchus-Early Olenekian, Ochev and Shishkin 2000) and Madagascar (Tertremoides-Olenekian unprecise age, Maganuco and Pasini 2009; Wantzosaurus-Olenekian unprecise age,

Acknowledgements Our special thanks to Dr. Rainer Schoch (SMNS, Stuttgart) for the access and loan of the studied specimen and to Dr. Igor V. Novikov (PIN, Moscow) for providing photos and relevant data about Russian trematosaurids. Sergio Llacer is acknowledged for his help during segmentation process of the studied specimen and Montse Vilalta for assistance in some drawings. Judit Marigó proofread the paper and revised English grammar. J. Fortuny acknowledges the support of the postdoc Grant Agència de Gestió d'Ajuts Universitaris i de Recerca, "Beatriu de Pinós" 2014-BP-A 00048 from the Generalitat de Catalunya. This work received support from CERCA programme (Generalitat de Catalunya).

Affiliations

Meritxell Fernández-Coll


Alain Pecker
email: alain.pecker@geodynamique.com

An estimate of maximum ground surface motion

Keywords: Soils, Earthquake, Wave propagation, Soil strength Mots-clés : Sols, Séisme, Propagation d'ondes, Résistance du sol

come

An estimate of maximum ground surface motion

Version française abrégée

La demande croissante d'études probabilistes de l'aléa sismique pour les installations essentielles pour la sûreté conduit à considérer des scénarios sismiques présentant des probabilités d'occurrence de l'ordre de 10 -7 par an donnant parfois naissance à des mouvements très élevés, qui calculés sur la base d'extrapolations de lois d'atténuation statistiques, peuvent atteindre plusieurs g. D'un point de vue physique ces mouvements ne peuvent exister car la capacité de résistance limitée des couches de sol, que traversent les ondes sismiques, limite les mouvements susceptibles d'être transmis. Un modèle analytique simple est proposé pour estimer l'accélération

Introduction

The increasing use of Probabilistic Seismic Hazard Assessment (PSHA) for critical facilities leads to the consideration of earthquake scenarios with a probability of occurrence as low as 10 -7 per year. For such low probabilities the computed ground accelerations, based on extrapolations of statistical attenuation relationships, may reach values as high as a few g's, which pose tremendous difficulties for earthquake resistance design. Obviously such large motions cannot exist because the soil profile, through which the seismic waves travel to reach the ground surface, has a limited resistance capacity and cannot transmit any motion. When failure is reached at any depth within the soil profile, the incident motion is filtered and no motion larger than the motion reached at that stage can be transmitted to the upper strata. The soil acts as a fuse to safely limit the maximum ground surface acceleration. It is therefore necessary to develop an approach that takes into account the soil resistance to derive an upper bound estimate of the ground surface motion. Some authors have tentatively developed such an approach (Betbeder Matibet [START_REF] Matibet | Calcul de l'effet de site pour une couche de sol[END_REF]), but their solution is strongly dependent on the soil stress-strain curve and the associated parameters, which are difficult parameters to measure and to reliably assess. The approach developed herein aims at circumventing those difficulties; it is based on an analytical solution to the wave equation in an inhomogeneous soil profile and provides an estimate of the maximum ground surface acceleration. It is realized that the maximum ground surface acceleration is not the sole meaningful parameter, and that response spectra would represent a more complete description of the surface motion, but, nevertheless, it is believed that the results presented below are useful for PSHA.

Equation of motion

Let us consider a soil layer of finite thickness overlying a stiff bedrock, which, for the purpose of this study, will be considered as a rigid boundary. In a first step, the soil is assumed to be isotropic elastic, with a shear wave velocity increasing with depth according to some power law:

V (z) = V s z h p/2 ( 1
)
where h is the layer thickness and V s the shear wave velocity at depth h. The wave equation for a plane vertically incident shear wave can be written:

∂ 2 u ∂z 2 + p z ∂u ∂z - 1 V 2 s h z p ∂ 2 u ∂t 2 = 1 V 2 s h z p d 2 v g dt 2 (2)
where u is the horizontal displacement, relative to the bedrock, and v g the bedrock motion. The boundary conditions express that the relative displacement at the bedrock interface and the shear stress at the ground surface are equal to 0:

z = 0, lim z→0 G(z) ∂u ∂z = 0 ( 3 a ) z = h, u = 0 (3b)
In Eq. (3a) G(z) is the shear modulus related to the shear wave velocity by the relationship G(z) = ρV 2 (z), where ρ is the soil mass density, assumed constant over depth.

Eq. ( 2) is solved with the mode superposition technique. The modes equation can be written:

d 2 X dz 2 + p z dX dz + ω 2 V 2 s h z p X = 0 ( 4 )
and its solution is given by (p < 2):

X = Az (1-p)/2 J (p-1)/(2-p) 2ωh p/2 V s (2 -p) z (2-p)/2
(5)

where J ν (•) is the Bessel function of order ν. A similar expression has been obtained by Dakoulas and Gazetas [START_REF] Dakoulas | A class of inhomogeneous shear models for seismic response of dams and embankments[END_REF] for the seismic response of earthdams.

Noting that (Abramowitz and Stegun [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]):

lim z→0 J ν (z) = z 2 ν 1 (ν + 1) + O z ν+2 (6)
where is the Gamma function, it is checked that the surface boundary condition is satisfied. The other boundary condition at z = h gives the frequency equation:

X(h) = 0 ⇒ J (p-1)/(2-p) 2ωh V s (2 -p) = 0 ( 7 )
The eigenfrequencies are given by:

f i = ω i 2π = λ i V s (2 -p) 4πh (8)
where λ i is the i-th root of the Bessel function:

J (p-1)/(2-p) (λ i ) = 0 (9)
Finally for convenience the mode shapes are normalized to unity at the ground surface X i (0) = 1; the final solution for the mode shapes is given, in dimensionless form, by:

X i z h = 2 λ i (p-1)/(2-p) 1 2 -p z h (1-p)/2 J (p-1)/(2-p) λ i z h (2-p)/2 (10)
For a constant mass density with depth, the mode participation factors, α i , are:

α i = 1 0 X i (u) du 1 0 X 2 i (u) du (11)
A closed form solution can be derived for the mode participation factor, but its numerical evaluation is not easy. Except for the special case p = 0, it is better to compute α i numerically from Eq. (11).

Finally the maximum displacement response in each mode is given by:

u i (z) = α i S d (ω i , ξ i )X i z h ( 12
)
where S d (ω i , ξ i ) is the spectral relative displacement of the input motion vg for frequency ω i and damping ratio ξ i .

The maximum ground surface acceleration, which is close to the pseudo acceleration, due to the contribution of the first N modes is: where S a (ω i , ξ i ) = ω 2 i S d (ω i , ξ i ) is the spectral pseudo acceleration. Eq. ( 13) takes into account the fact that the mode shapes have been normalized to 1 at the ground surface (X i (0) = 1). With four modes the cumulative modal mass is greater than 90% (this property is even satisfied with three modes provided p 0.75) and therefore the analysis can be restricted to the first three or four modes (N = 3 or 4). The roots λ i of the frequency Eq. ( 9) and the mode participation factor α i for the first four modes are given in Table 1 for different values of the exponent p in Eq. ( 1).

ümax (z = 0) = N i=1 α i S a (ω i , ξ i ) 2 1/2 (13)
The shear strain is given, for each mode, by:

γ i (z) = ∂u i (z) ∂z = α i S a (ω i , ξ i ) hω 2 i dX i (u) du (14)
Given the relationship:

J ν (x) = J ν-1 (x) - ν x J ν (x) ( 15
)
the shear strain can be normalized to yield a quantity that is independent of the input motion and only depends on the soil profile through the exponent p:

χ i z h = γ i (z)ω 2 i S a ω i , ξ i h = α i 2 λ i ν 1 2 -p × (1 -p) z h -(p+1)/2 J ν λ i z h (2-p)/2 + λ i 2 -p 2 z h (1-2p)/2 J ν-1 λ i z h (2-p)/2 (16) with ν = (p -1)/(2 -p).
When z tends to zero the normalized shear strain is equivalent to:

χ i z h ∼ -α i (2 -p) 2 λ i 2 2 z h 1-p (17)
It can be checked that the shear stress, which is equal to the shear strain times the shear modulus, tends to zero because G is proportional to z p . The normalized shear strain can be determined once and for all. Sample plots are shown in Fig. 1 for p = 0 and p = 0.75.

Soil constitutive model

A simplified elastic-perfectly plastic constitutive relationship (Fig. 2) is assumed for the soil layer; the shear stress-shear strain curve is defined by two parameters which may depend on the depth z:

• the shear strength τ max (z); • the yield strain γ f (z). The shear modulus is then given by G = τ max /γ f . As soon as, at any depth within the soil profile, the shear strain reaches γ f , the maximum shear stress that can be transmitted is limited by τ max ; the ground surface acceleration cannot therefore exceed the value reached when γ (z) = γ f .

The procedure involves the following steps:

(i) define the input motion at the rock interface by its pseudo acceleration response spectrum S * a . This is typically the result of the PSHA; (ii) compute the eigenfrequencies and mode participation factors from Eqs. ( 8) and ( 11); (iii) plot the shear strain (Eq. ( 16)) versus depth together with the yield strain γ f ; (iv) determine the depth z 0 and the scaling factor µ for which γ f = µγ (z 0 ); (v) define S a = µS * a the maximum possible pseudo acceleration from which the maximum ground surface acceleration ümax (z = 0) is determined (Eq. ( 13)).

A table and a figure are provided for the relevant input data for the procedure defined above:

• Table 1 gives the mode participation factors α i and the roots of the Bessel function for the first three modes, to be used for step (ii); • Fig. 1 gives the variation of the normalized shear strain versus depth χ(z/h) for the first four modes and different values of the parameter p, to be used for step (iii).

Example of application

Let us consider a site composed of 50 m of sand and gravel with a water table at 26 m depth. The shear strength is expressed by the Mohr Coulomb strength criterion τ max = σ v tan(ϕ ) with a friction angle ϕ = 41 • . The yield strength is equal to 3%. From the constitutive relationship the variation of the shear wave velocity with depth can be approximated with p = 0.75 and V s = 104 m/s. The bedrock motion is represented by the pseudo acceleration response spectrum of Fig. 3, scaled to 10 m/s 2 .

The first three circular frequencies, which are required to achieve a modal mass greater than 90%, are equal to: ω 1 = 2.72 rd/s, ω 2 = 6.78 rd/s, ω 3 = 10.87 rd/s and the corresponding spectral accelerations for 20% damping, which is relevant for a near failure condition, are equal to S * a1 = 2.37 m/s 2 , S * a2 = 6.83 m/s 2 , S * a3 = 10.94 m/s 2 . The yield strain becomes equal to the induced strain for µ = 1.78 at a critical depth of 3 m below the ground surface.

The associated spectral accelerations are: S a1 = µS * a1 = 4.22 m/s 2 , S a2 = µS * a2 = 12.16 m/s 2 , S a3 = µS * a3 = 19.47 m/s 2 from which the maximum peak ground acceleration (Eq. ( 13)) is equal to: ümax = 15.7 m/s 2 .

Conclusions

A simple method has been proposed to estimate the maximum ground surface motion that can reach the surface of a soil profile whatever the amplitude of the input rock acceleration. Tables and figures are provided that makes the method readily applicable to any soil profile and input motion. The proposed method is robust, because the shear strength is a parameter that is routinely measured, rather reliable, and also because the results are not sensitive to changes in the yield strain. For the example presented herein it has been compared to parametric numerical site response analyses based on a sophisticated soil constitutive model (multi-yield elastoplastic model with kinematic hardening) from which the maximum ground surface acceleration has been found equal to 12 m/s 2 . When compared with actual earthquake records (Rosrine SMT site in California) it does not contradict the observations: the maximum recorded acceleration is equal to 8.9 m/s 2 whereas the predicted one using the above methodology is equal to 8.8 m/s 2 .

Fig. 1 .

 1 Fig. 1. Normalized shear strain versus normalized depth.

Fig. 1 .

 1 Fig. 1. Déformation de cisaillement normalisée en fonction de la profondeur normalisée.

Fig. 2 .

 2 Fig. 2. Soil stress-strain behavior.

Fig. 2 .

 2 Fig. 2. Loi de comportement du sol.

Fig. 3 .

 3 Fig. 3. Bedrock response spectrum.

Fig. 3 .

 3 Fig. 3. Spectre de réponse au rocher.

Table 1

 1 Roots of Bessel equation and mode participation factors

	Tableau 1
	Racines de l'équation de Bessel et coefficients de participation modaux
	p		0.0	0.25	0.5	0.75	1.0
	Mode 1	λ	1.5708	1.7004	1.8664	2.0883	2.4048
		α	1.2732	1.3165	1.3760	1.4630	1.6020
	Mode 2	λ	4.7124	4.8313	4.9879	5.2035	5.5201
		α	-0.4244	-0.5010	-0.6104	-0.7781	-1.0648
	Mode 3	λ	7.8540	7.9703	8.1243	8.3377	8.6537
		α	0.2546	0.3149	0.4068	0.5600	0.8514
	Mode 4	λ	10.9956	11.1107	11.2635	11.4759	11.7915
		α	-0.1819	-0.2314	-0.3099	-0.4480	-0.7297