Ronald Noë

Marion Laporte

Socio-spatial cognition in vervet monkeys

Keywords: spatial cognition, social brain, group cohesion, field experiment, vervet monkey, Chlorocebus pygerythrus

Safety-in-numbers is thought to be the principal advantage of living in groups for many species. The group can only provide protection against predators, however, when group cohesion is maintained. Vocalisations are used to monitor inter-individual distances, especially under conditions of poor visibility, but should be avoided in the presence of predators. Mentally tracking the movements of silent and invisible group members would allow animals foraging in dense vegetation to stay close to their group members while reducing the use of vocal contact. We tested the socio-spatial cognitive abilities of wild vervet monkeys (Chlorocebus pygerythrus) by comparing their reactions to plausible and implausible displacements of group members simulated by sound playbacks. Our methods are comparable to those used in studies of 'object permanence' and 'invisible displacements' of inanimate objects. Our results show that vervets can track the whereabouts of invisibly and silently moving group members, at least over short periods of time.

Introduction

'The social brain' is shorthand for a now widely accepted explanation of the evolution of advanced cognitive abilities in animals and especially primates. The social brain hypothesis states that group living has selected for complex cognitive capacities, resulting in an increase in brain size, notably the neocortex [START_REF] Dunbar | Neocortex size as a constraint on group size in primates[END_REF][START_REF] Dunbar | The social brain hypothesis[END_REF][START_REF] Dunbar | Evolution of the social brain[END_REF][START_REF] Humphrey | The social function of intellect[END_REF]; [START_REF] Whiten | Tactical deception in primates[END_REF]. This explanation provides an alternative to the idea that advanced abilities evolved in reaction to ecological challenges [START_REF] Clutton-Brock | Primates, brains and ecology[END_REF][START_REF] Harvey | Brain size, energetics, ecology and life-history patterns[END_REF][START_REF] Sawaguchi | The size of the neocortex in relation to ecology and social structure in monkeys and apes[END_REF]. The tendency to seek the company of conspecifics, in turn, is thought to have evolved under predation pressure according to the 'socio-ecological hypothesis' [START_REF] Sterck | The evolution of female social relationships in nonhuman primates[END_REF][START_REF] Van Schaik | Why are diurnal primates living in groups?[END_REF][START_REF] Van Schaik | On the ultimate causes of primate social systems[END_REF]. Group-living comes at a cost, however, notably in the form of increased competition for food and mates [START_REF] Cheney | The influence of intergroup competition on the survival and reproduction of female vervet monkeys[END_REF][START_REF] Dunbar | The social brain hypothesis[END_REF][START_REF] Van Schaik | Why are diurnal primates living in groups?[END_REF], but also as an increased risk of contracting infectious diseases (Altizer et al. 2003 and references therein;[START_REF] Nunn | Emerging infectious diseases and animal social systems[END_REF]. Cognitive challenges specific to group-living animals include keeping track of the interactions with group members, of relationships between third parties, of changes in dominance relationships, forming alliances as well as monitoring alliances formed by others and so forth [START_REF] Bergman | Hierarchical classification by rank and kinship in baboons[END_REF][START_REF] Cheney | How monkeys see the world[END_REF][START_REF] Dasser | A social concept in Java monkeys[END_REF][START_REF] Dunbar | Evolution in the social brain[END_REF][START_REF] Range | Can simple rules account for the pattern of triadic interactions in juvenile and adult female sooty mangabeys?[END_REF][START_REF] Shultz | The evolution of the social brain: anthropoid primates contrast with other vertebrates[END_REF][START_REF] Shultz | Stepwise evolution of stable sociality in primates[END_REF]. One cognitive challenge directly related to the 'safety-in-numbers' function of sociality is seldom discussed, however: the necessity to keep track of the spatial positions of group members even under conditions of poor visibility as found in dense forests and scrublands and especially when vocalisations increase the individual predation risk. We propose the label "socio-spatial cognition" for this cognitive capacity. This label suggests a cognitive ability linked to group living, but it is unlikely to be limited to species living in permanent groups. For most animals the ability of tracking the whereabouts of living things, be it conspecifics, prey or predators, will have positive effects on fitness. We assume, however, that the complexity of simultaneously tracking multiple group members selected for additional proficiency.

With the exception of some terrestrial species, most primate groups forage and travel in relatively dense environments. Groups tend to remain coherent in spite of the fact that Noë & Laporte Socio-spatial Ms.docx 3/27 some members cannot see each other over longer periods of time. Coherence can in principle be maintained by three complementary mechanisms: (1) each group member remains in visual contact with at least one other member and a network of visual contacts connects all members of the group; (2) group members are in regular auditory contact with each other, notably by emitting soft vocalisations, usually labelled 'contact calls', and (3) each individual has a mental representation of the location of the members of his group, or at least those in its direct vicinity, even when these move and are silent and out of sight. In this paper we focus on this last and most cognitively challenging aspect of socio-spatial cognition: the ability to mentally represent group members that (1) are out of sight and produce no sounds by which they can be recognized individually and (2) mentally project the path along which invisible and silent group members are likely to move during foraging or travel. The first requires a capacity for object permanence and the second for tracking invisible displacements of objects, both phenomena that were originally defined by [START_REF] Piaget | La construction du réel chez l'enfant. Delachaux & Niestlé[END_REF] in the context of human development, albeit that the objects are in our case living animals and that their movements can be much more complicated than the linear object displacements usually tested in experiments on spatial cognition. Relatively few studies used dynamic and animate objects (reviewed in [START_REF] Call | Representing space and objects in monkeys and apes[END_REF][START_REF] Shettleworth | Cognition, Evolution and Behavior[END_REF]). Exceptions include a study by Bates and colleagues (2008) in which elephants are reported to be able to recognise group members by scent and to track their positions while walking. Wich and De Vries (2006) showed that male langurs have a representation of the membership of their group, but not necessarily of the positions of individual members. In the presence of a predator, the langurs only stopped giving alarm calls when all the individuals had reacted by giving at least one call.

Ecological spatial orientation

Taking direct routes between different locations in the home range towards goals that are initially not visible has been shown for many primate species, both in the wild (e.g. [START_REF] Boesch | Mental map in wild chimpanzees: an analysis of hammer transports for nut cracking[END_REF][START_REF] Garber | Role of spatial memory in primate foraging patterns: Saguinus mystax and Saguinus fuscicollis[END_REF][START_REF] Sigg | Home range and daily march in a Hamadryas baboon troop[END_REF] and in captivity [START_REF] Ludvig | Spatial memory performance of freelymoving squirrel monkeys[END_REF]. For example, hamadryas baboons use a least distance strategy and apparently know that they are approaching their goal before seeing it, accelerating their Noë & Laporte Socio-spatial Ms.docx 4/27 pace in anticipation [START_REF] Sigg | Home range and daily march in a Hamadryas baboon troop[END_REF]. Chimpanzees remembered up to 18 different locations during experiments in large enclosures [START_REF] Menzel | Chimpanzee spatial memory organization[END_REF]. Vervet monkeys could only memorize six distinct food locations during similar experiments, but seemed to anticipate their path three steps ahead, thus solving a simple 'travelling salesman problem' [START_REF] Cramer | Vervet monkeys as travelling salesmen[END_REF]. Japanese monkeys made a beeline for akebi fruit at locations where akebi had been found in the previous fruiting season after finding ripe fruit placed on their path ahead of the season [START_REF] Menzel | Cognitive aspects of foraging in Japanese monkeys[END_REF].

Dear enemies

Vervet monkeys were also shown to possess socio-spatial cognition at a larger scale.

Using playback experiments Cheney and Seyfarth (1982b) could show that neighbouring groups were associated with vocalisations from specific directions. This ability to associate locations and individuals was first established in territorial birds and is usually referred to as the 'dear enemy' phenomenon [START_REF] Fisher | Evolution and bird sociality[END_REF], also known as the 'neighbour-stranger' effect. Territorial individuals respond with greater aggression to strangers than to familiar neighbours, implying that they can recognise their neighbours and associate them with a specific territory as was shown in a series of seminal papers by Brooks and Falls (Brooks and Falls 1975a, b;[START_REF] Falls | Individual recognition by song in white-throated sparrows .2. Effects of location[END_REF]. The phenomenon has been shown in numerous species [START_REF] Akçay | Good neighbour, bad neighbour: song sparrows retaliate against aggressive rivals[END_REF][START_REF] Briefer | When to be a dear enemy: flexible acoustic relationships of neighbouring skylarks, Alauda arvensis[END_REF][START_REF] Carazo | Beyond 'nasty neighbours' and 'dear enemies'? Individual recognition by scent marks in a lizard (Podarcis hispanica)[END_REF][START_REF] Leiser | The benefits of dear enemy recognition in three-contender convict cichlid (Cichlasoma nigrofasciatum) contests[END_REF]Temeles 1994 and references therein;[START_REF] Zenuto | Dear enemy relationships in the subterranean rodent Ctenomys talarum: the role of memory of familiar odours[END_REF], including several primate species (Kitchen and Beehner 2007 and references therein;[START_REF] Thompson | Why fight? Selective forces favoring betweengroup aggression in a variably pair-living primate, the white-faced saki (Pithecia pithecia)[END_REF].

Invisible displacements

The Piagetian theory on object permanence, including visible and invisible displacements, suggests how children mentally represent out-of-sight objects [START_REF] Piaget | La construction du réel chez l'enfant. Delachaux & Niestlé[END_REF]Piaget and Inhelder 1966). Object permanence and an understanding of visible displacements has been demonstrated in a wide range of vertebrate species, for example primates [START_REF] De Blois | Object permanence in orangutans (Pongo pygmaeus) and squirrel monkeys (Saimiri sciureus)[END_REF][START_REF] De Bois | Object permanence in rhesus monkeys (Macaca mulatta)[END_REF][START_REF] Deppe | Object permanence in lemurs[END_REF];

Noë & Laporte Socio-spatial Ms.docx 5/27 [START_REF] Mendes | Object permanence in common marmosets (Callithrix jacchus)[END_REF], domesticated carnivores [START_REF] Fiset | Duration of cats' (Felis catus) working memory for disappearing objects[END_REF][START_REF] Fiset | Invisible displacement understanding in domestic dogs (Canis familiaris): the role of visual cues in search behavior[END_REF][START_REF] Gagnon | Search behavior of dogs (Canis familiaris) in invisible displacement problems[END_REF], dolphins [START_REF] Jaakkola | What do dolphins (Tursiops truncatus) understand about hidden objects?[END_REF]) and birds [START_REF] Bugnyar | The ontogeny of caching in ravens, Corvus corax[END_REF]). The ability to correctly locate invisibly moving objects has, however, only been reported in psittacine birds [START_REF] Pepperberg | Development of Piagetian object permanence in grey parrot (Psittacus erithacus)[END_REF], corvids [START_REF] Bugnyar | The ontogeny of caching in ravens, Corvus corax[END_REF][START_REF] Pollok | Development of object permanence in food-storing magpies (Pica pica)[END_REF], domesticated carnivores [START_REF] Collier-Baker | Do dogs (Canis familiaris) understand invisible displacement?[END_REF][START_REF] Dumas | Object permanence in cats (Felis catus) -an ecological approach to the study of invisible displacements[END_REF], monkeys [START_REF] Mendes | Object permanence in common marmosets (Callithrix jacchus)[END_REF][START_REF] Neiworth | A test of object permanence in a new-world monkey species, cotton top tamarins (Saguinus oedipus)[END_REF] and great apes [START_REF] Albiach-Serrano | Great apes track hidden objects after changes in the objects' position and in subject's orientation[END_REF][START_REF] Barth | Tracking the displacement of objects: a series of tasks with great apes (Pan troglodytes, Pan paniscus, Gorilla gorilla, and Pongo pygmaeus) and young children (Homo sapiens)[END_REF][START_REF] Collier-Baker | Do chimpanzees (Pan troglodytes) understand single invisible displacement?[END_REF][START_REF] De Blois | Object permanence in orangutans (Pongo pygmaeus) and squirrel monkeys (Saimiri sciureus)[END_REF], whereby the results of several studies fail to lend unequivocal support to the presence of this cognitive ability in the species studied. Predicting the future positions of moving targets on computer screens, as shown in studies using eyemovements (Ferrera and Barborica 2010 and references therein) is a related skill, but its relevance to movements at the scale considered in this study is less obvious.

Testing socio-spatial cognition in vervets

We tested whether vervet monkeys can mentally follow invisible displacements of silent group members under natural conditions using playbacks of the most commonly uttered vocalisation, the contact grunt. Contact grunts are short range affiliative calls emitted in different contexts: when encountering another member of the group, dominant or subordinate, when moving to an open area, or when another group has just been spotted, as described, among others, by Cheney and Seyfarth (1982a) for vervets and by Rendall and colleagues for baboons [START_REF] Rendall | The meaning and function of grunt variants in baboons[END_REF]. Although these grunts tend to show acoustic differences in different contexts, the ones selected for the playbacks were not emitted, during agonistic interactions, during encounters with other groups, or after other strong stimuli of any kind that we were aware of. We assumed the main function of these grunts to be providing information of the whereabouts of the caller to its fellow group members and perhaps also eliciting similar calls from those group members (Fig. 1 gives an example of the calls used).

---Figure 1 about here ---

Noë & Laporte Socio-spatial Ms.docx 6/27
We tested two predictions using methods illustrated in Fig. 2. Prediction 1: vervet monkeys can mentally follow the trajectories of invisible group members, at least over short time spans. In experiment A (Fig. 2A) we played the contact call of an out of sight individual (emitter) to the subject from an 'impossible location', i.e. a location that the individual could not have reached given the distance and available time (test condition) or from a 'possible' location (control condition).

Prediction 2: vervets have a mental representation of the natural movement of other vervets, i.e. they can predict the trajectories of out-of-sight group members based on their experience with their usual speed during specific group activities, notably foraging and travelling, perhaps even taking the local topography into account. In experiment B (Fig. 2B), we played two contact calls of an individual that was out-of-sight during the entire trial (emitter) to the subject, simulating either an impossible (test condition) or a possible (control condition) movement.

--Fig. 2 about here --

Methods

Study site and subjects

Subject were members of two non-neighbouring groups of vervet monkeys (Chlorocebus pygerythrus) in the Loskop Dam Nature Reserve, Mpumalanga, South Africa. The reserve, situated 250 km north east of Johannesburg (-25° 26' 31.69", +29° 15' 19.14"), covers 25 000 ha of bushveld (a mixture of tall grasses and thick acacia bushes). The Donga group had a home range of 224 ha and consisted of 15 individuals and the Picnic group had range of 138 ha and 11 members (for a detailed description see: [START_REF] Barrett | A floristic description and utilisation of two home ranges by vervet monkeys in Loskop Dam Nature Reserve, South Africa[END_REF]). These ranges were separated by about 3 km. Our groups, like most vervet groups, contained multiple adult males and females. Females usually remain in their natal group and form stable hierarchies organised in matrilines, while most males migrate to other groups before reaching sexually maturity. During the habituation period from January till May, data on social behaviour were collected using mainly ad libitum and focal-animal sampling methods [START_REF] Altmann | Observational study of behavior: sampling methods[END_REF]. This allowed us to ). We tried to be as close to the emitter as possible (between 3 and 8 meters) without disturbing the animals in their activities. Vocalisations of 13 members of the two study groups were used (details in Table 2). In addition, we used the voices of two females, one adult and one juvenile, recorded at the Vervet Monkey Foundation at Tzaneen, Limpopo Province, South Africa as vocalisations of 'strangers'. Problems with unnatural background noises prevented further use of recordings made at Tzaneen.

Experiments

Playback trials were done by ML using a Sony recorder (DAT TCD-D8) connected to a 12V Pioneer amplifier working on a small battery and connected to one or two Bose 151 loudspeakers, depending on the type of trial. During the trials the subjects were filmed with a Panasonic digital video camera (NV-GS11) from at least 5 seconds before till 60 seconds after each playback.

Individual recognition by voice is a general phenomenon in animals [START_REF] Tibbetts | Individual recognition: it is good to be different[END_REF] and has been shown a long time ago in vervets (Cheney andSeyfarth 1980, 1982b).

Nevertheless we verified this in a preliminary experiment that also served to test our equipment, by playing the contact calls recorded at the Tzaneen Vervet foundation to both our study groups as well as calls recorded in one of our study groups to members of the other one. As controls we used all trials with playbacks from 'possible locations' of the two experiments A and B (AC, BC1 and BT1 as described below), i.e. all trials with familiar voices played from plausible locations.

For experiment A (Impossible location -Fig 2a ), we hid the loudspeaker as soon as a favourable situation presented itself: with at least two individuals well visible and the group resting or foraging. We waited till an individual left the field of vision of the vervet left behind and then played back the contact call of the animal that had disappeared within 2 min at about 5 meters from the subject tested. The trial was considered as a control (AC) when the moving animal had disappeared from view within 45° at either side of the loud speaker. All other directions of movement were considered as a test (AT) conditions. The subject was expected to show a sign of surprise in AT-trials, but not in AC-trials.

For experiment B (Unlikely movements -Fig. 2b), two different calls of the same individual that followed each other in the original recording were played from two different loudspeakers with a delay of 5-10 sec. The animal whose voice was used was out-of -sight of the subject tested from the moment the loudspeakers were put in position till the end of the trial. The loudspeakers emitting the first calls in both the control (BC1) and the test condition (BT1) were always in a position concordant with the emitter's last observed position, as in the AC-trials of experiment A. The second call in the control condition (BC2) came from a second loudspeaker placed at less than 1 m from the first. Only the second call in the test condition (BT2), which came from a location at 7 to 10 m (depending on the local terrain) from the first and from opposite direction of both the position of the animal whose voice was used and of the loudspeaker emitting the first call (BT1), was expected to produce a strong reaction of surprise.

---Figure 2 (a) and (b) about here ---A trial was considered as valid if no unusual event disturbed it, such as a fight, an alarm call, or other event that could change the receiver's attitude and if no vocalisations by any member of the group could be heard shortly before or during the trial. In all those cases, the trial was aborted and no further attempts were made that same day. In order

Noë & Laporte Socio-spatial Ms.docx 9/27 to avoid the habituation of monkeys to the experimental procedure no individual was used as test subject in the same experiment more than once a day. Due to the low probability that all the favourable conditions were met, valid trials were infrequent (on average once every three days in each group). We had to use a few subjects in multiple trials (Table 2), but otherwise pseudo-reciprocity was avoided as much as possible by using both vocalisations of different individuals and different vocalisations of the same individual.

---Table 2 about here ---

Video analysis

A quick and radical change of behaviour immediately after the playback served as our main criterion for the presence or absence of a reaction associated with 'surprise', shorthand for a reaction to an unusual and/or unexpected event. The end of the reaction period was defined by a return to the initial activity or another routine activity for more than five seconds. The videos were analysed by ML and four naive observers that were all experienced observers of vervet behaviour, but did not know whether they looked at a test or control trial. The videos were muted, except for a 'beep' when the vocalisation was played and were analysed in random order. We used both qualitative and quantitative variables: we scored the degree of reaction and the instantaneity of the reaction and measured the duration of the reaction and the number of head rotations (Table 3). Kendall's tau-b between ML and the observers: EW: 0.783; CF: 0.664; AB: 0.659; YG: 0.775. In addition we asked YG to measure the instantaneity of the reaction (Cohen's kappa: 0.886) and the number of head rotations (Pearson r: 0.996). On the basis of the high level of agreement we decided to use ML's original scores for the analysis.

---Table 3 ---

Statistical analysis

Noë & Laporte Socio-spatial Ms.docx 10/27

The differences between test and control conditions were examined at the individual level using Mann-Whitney U and Wilcoxon tests for quantitative variables and Fisher tests for qualitative variables [START_REF] Siegel | Nonparametric statistics for the behavioral sciences[END_REF]. For individuals tested more than once we used the average score for quantitative variables and the result of the first trial for qualitative variables. The alpha-level was set at 0.05, but whenever datasets were used in two different tests, a Bonferroni corrected alpha of 0.025 was used. Exact p-values were calculated for small sample sizes [START_REF] Mundry | Use of statistical programs for nonparametric tests of small samples often leads to incorrect P values: examples from Animal Behaviour[END_REF]. Tests are two-tailed unless indicated otherwise. Data analysis was carried out with SPSS 18.0.

The AC trials were difficult to perform and therefore only a few of them were done with different subjects. We checked if all trials from the different experiments in which the playback came from a 'possible location' could be pooled together by performing a Kruskal Wallis test, since these trials are de facto identical in design . We only used the reactions to the first playbacks in the experiment B (with two loudspeakers) to prevent a possible effect of a redundant stimulus. The tests revealed no significant differences across conditions for the number of head rotations (H(2) = 3.55, p = 0.169) and the reaction length (H(2) = 3.79, p = 0.150). We therefore decided to pool these data to supplement our analyses when needed.

Results

Preliminary experiment: group member -stranger discrimination

We compared reactions to the playbacks of strangers with reactions to the playbacks of group members broadcasted from a possible location. When hearing the vocalisation of a stranger, the vervets showed a significantly stronger reaction (Fisher's exact test: N = 9, p = 0.016 and with the controls pooled N = 22, p = 0.029), responded faster (Fisher's exact test: N = 9, p = 0.047 and with the controls pooled N = 22, p = 0.054), with more head rotations (Mann-Whitney test: one-tailed, N stranger = 5, N controls pooled = 11, U = 0, z = -3.17, pexact = 0.0002) and it took longer before they returned to their daily routine (Mann-Whitney test: one-tailed, N stranger = 5, N controls pooled = 11, U = 0, z = -3.14, pexact = 0.0002). = -2.74, pexact = 0.002; with the controls pooled, one-tailed, U = 2, z = -3.49, pexact = 5.3x10-3) and the number of head rotations was higher (Mann-Whitney test: one-tailed, U = 0.5, z = -2.65, pexact = 0.004; with the controls pooled, one-tailed, U = 6.5, z = -3.13, pexact = 5.3x10 -3 ). For three subjects we had matched test and control trials in which the voice of the same individuals were used. The subject's reaction was stronger for all parameters in the test trials compared to the control trials.

---Figure 4 about here ---

Experiment B: unlikely movements

We first tested the reaction to an impossibly fast movement of an invisible individual by comparing the reactions to the sound of the first loudspeaker (BT1) with the reactions to the sound coming from the opposite direction (BT2) for eight individuals (with the emitters of the voice used always out-of-sight). The sounds coming from the opposite direction elicited reactions that were stronger (Fisher's exact test: N = 16, p = 0.014), but not faster (Fisher's exact test: N = 16, p = 0.47), lasted longer (Wilcoxon signed rank-test: one-tailed, Z = -2.52, N = 8, pexact = 0.0039) and were followed by a higher the number of head rotations (Wilcoxon signed rank-test: one-tailed, Z = -2.52, N = 8, pexact = 0.0039). As a control, we confronted five different subjects with two different

Noë & Laporte Socio-spatial Ms.docx 12/27 recordings (BC1 and BC2) of vocalisations of a same individual emitted at the same interval as used in the test trials from two loudspeakers that were placed close to each other and were both at a plausible location. The degree of reaction to the two sounds was not significantly different based on the judgement of surprise (Fisher's exact test: N = 10, p = 1), the instantaneity of the reaction (Fisher's exact test: N = 10, p = 1), the total time before returning to routine behaviour (Wilcoxon signed rank-test: one-tailed, Z = -0.73, N = 8, pexact = 0.31) and the number of head rotations (Wilcoxon signed rank-test: one-tailed, Z = -0.55, N = 8, pexact = 0.37).

Finally, we compared the second playbacks from the test (BT2) and control (BC2) trials.

The degree of reaction to the test trials only showed an insignificant trend of being stronger (Fisher's exact test: N = 13 p = 0.084). However, the reaction was significantly faster (Fisher's exact test: N = 13, p = 0.007), lasted longer (Mann Whitney test: onetailed, U = 4, z = -2.35, pexact = 0.0093) and the number of head rotations was higher (Wilcoxon signed rank-test: one-tailed, U = 6.4, z = -2.00, N = 8, pexact = 0.021, Bonferroni corrected α = 0.025). As in experiment A, we could also compare the results for test and control trials with the same voice for three subjects. The results were in line with the complete set.

---Figure 5 about here ---

Discussion

Following out-of-sight movements

In experiment A, we played the emitter's voice to the subject after the emitter had left the subject's visual field. During test trials in which a vocalisation was played back from a different direction than the one in which the emitter had left, we observed a significantly stronger reaction than in control trials in which we played the emitter's voice from the direction in which (s)he moved out-of-sight. This was also true for trials in which we used the vocalisation of only one of several individuals that had just disappeared from sight. Subjects generally showed little or no reaction to the playbacks during control trials. However, they sometimes replied to the playback with a contact

Noë & Laporte Socio-spatial Ms.docx 13/27 call, but without looking up and without showing a surprised reaction. This suggests to us that the vervets considered this situation as a routine event. The subject could only have been updated about the emitter's real position by movements of the vegetation and non-vocal sounds caused by the emitter's movements, but such clues do not allow individual recognition and thus do not invalidate a test of the ability of tracking invisible displacements. We therefore conclude that the subjects expectations were violated, which suggests that vervets are able to track the movements of group members at least for a few minutes after they disappear from sight. This is in agreement with studies on target tracking by monkeys [START_REF] Ferrera | Internally generated error signals in monkey frontal eye field during an inferred motion task[END_REF][START_REF] Filion | Can monkeys (Macaca mulatta) represent invisible displacement?[END_REF][START_REF] Neiworth | A test of object permanence in a new-world monkey species, cotton top tamarins (Saguinus oedipus)[END_REF][START_REF] Washburn | Comparative assessment of psychomotor performance: target prediction by humans and macaques (Macaca mulatta)[END_REF], but in contradiction with some studies on invisible displacement suggesting that monkeys cannot represent out of sight movements [START_REF] Call | Representing space and objects in monkeys and apes[END_REF][START_REF] De Blois | Object permanence in orangutans (Pongo pygmaeus) and squirrel monkeys (Saimiri sciureus)[END_REF][START_REF] De Bois | Object permanence in rhesus monkeys (Macaca mulatta)[END_REF][START_REF] Gomez | Species comparative studies and cognitive development[END_REF]Natale et al. 1986 and references therein).

Implausible speed

In experiment B, contact calls were played back from two directions, diametrically opposed from the point of view of the receiver (test condition) or from two locations very close to one another (control condition). The movements suggested by the test trials were highly implausible, because a real monkey would have had to run through the open area around the subject, or run extremely fast behind the vegetation surrounding the subject, to get from one speaker position to the next within this time frame. The subjects clearly showed reactions of surprise in the test trials but not in the controls. No difference between the first and the second playback (BT1 and BT2) was found for the parameter 'instantaneity of the reaction'. This can be due to the binary character of the parameter used (reaction within a second or not) combined with a small sample size. However, the reactions were overall still stronger, lasted longer, and involved more head rotations. We conclude that the subjects' expectations about the speed with which conspecifics can move from one location to the next were violated. This goes beyond the results of [START_REF] Hauser | A nonhuman primate's expectations about object motion and destination: The importance of self-propelled movement and animacy[END_REF], who showed in a lab experiment that tamarins expect animals to move to other locations when out of sight, but not inanimate objects.

Noë & Laporte Socio-spatial Ms.docx 14/27 A less cognitively demanding hypothesis would be that the subjects do not represent the movements of group members but know that an individual cannot occupy two positions at the same time. In contrast to the experiment A, in which the departure of the emitter indicated a movement in a particular direction, the two vocalisations of experiment B only gave indications about two static locations. We cannot exclude that this just represents a violation of expectations about the same individual being at two distant locations rather than about movement. Nevertheless, we feel that 'implausible speed' and 'unlikely movement' are appropriate terms. Our reasoning is as follows: The simplest explanation for a surprise reaction after hearing the same voice from two different locations is that it violates the day-to-day experience of the animals: the same voice can be heard from different locations, but not normally with such a short time interval, at least not without any sign of an individual moving fast. Hearing the same sound from different locations at short intervals does occur occasionally, however: vervets sometimes run from one spot to the next while vocalising and screaming infants tumble from trees at times. Such natural events always involve fast, and therefore rather conspicuous, displacements. Thus, what surprises them is less likely to be the fact that the two sounds come from different locations shortly after each other as such, but rather the lack of fast movements and sounds that normally accompany this.

A functional perspective

The question is whether a monkey tracking a group member's movement in the wild can be compared to tracking an object, animated or inanimate, in the lab. Loosing track of an object does not normally have any serious fitness consequences, but losing track of one's group members can have serious, even fatal, consequences. The two types of experiments are therefore hard to compare in terms of ecological validity. Tracking group members in dense environments can be essential during encounters with sit-andwait predators when all members of the group try to move away silently [START_REF] Zuberbühler | Leopard predation and primate evolution[END_REF]. It may even be used to keep track of the predators themselves. Sociospatial skills might also play a role in coordinating group movement [START_REF] King | Where next? Group coordination and collective decision making by primates[END_REF], in cooperative group defence against neighbouring groups [START_REF] Kitchen | Factors affecting individual participation in group-level aggression among non-human primates[END_REF][START_REF] Meunier | Participation in group defence: proximate factors affecting male behaviour in wild white-faced capuchins[END_REF], or in more complex coordinated actions such as 'co-Noë & Laporte Socio-spatial Ms.docx 15/27 operative' hunting [START_REF]Chimpanzee Hunting Behavior[END_REF]. These forms of coordination do not necessarily imply complex mental representations or shared internal processes [START_REF] Barrett | Social brains, simple minds: does social complexity really require cognitive complexity?[END_REF][START_REF] Knoblich | Action coordination in groups and individuals: Learning anticipatory control[END_REF]. Socio-spatial cognition can also be important for following of the positions of potential allies or competitors. Some authors noted that individuals adopt different strategies in competitive situations depending on the identity of individuals in their surroundings [START_REF] Janson | Social correlates of individual spatial choice in foraging groups of brown capuchin monkeys, Cebus apella[END_REF][START_REF] Robinson | Spatial structure in foraging groups of wedge-capped capuchin monkeys Cebus nigrivittatus[END_REF][START_REF] Seyfarth | Social awareness in monkeys[END_REF]. Finally, it might be useful for individuals that have an interest in withholding information. Chimpanzee females can, for example, refrain from giving a greeting grunt to a male when the alpha male is around [START_REF] Laporte | Vocal greeting behaviour in wild chimpanzee females[END_REF]. Experiments conducted by [START_REF] Seyfarth | Grooming, alliances and reciprocal altruism in Vervet monkeys[END_REF] suggested that vervets are more attentive to the movements of kin and individuals with which they recently had an affiliative interaction. We had the impression that our subjects kept track of all the individuals in their direct vicinity, independently of age, sex, rank or degree of kinship.

Experiment Voice owner Subject Tested

Group

Number of head rotations

Number of times the animal stops turning the head and gazes in a specific direction while scanning the environment.

Reaction length

Duration of head rotations until the individual goes back to his previous activity (feeding, grooming, etc) or disappears from sight. This was limited to 60 seconds of analysis because individuals then tend to move and it becomes very difficult to assess their behaviour without disrupting them.

  Noë & Laporte Socio-spatial Ms.docx 7/27 construct a hierarchy and document affiliative relations between individuals. The preliminary experiments were done in April and experiments A and B from June till September 2005. Recordings Vocalisations of individually recognised individuals were recorded by ML ad libitum with a Sennheiser directional microphone (ME 66) and a Sony digital recorder (DAT TCD D8

  test trials with eight different stationary individuals (3AF, 4AM and 1 JF) as subjects. As 'possible location' controls we used only the four control trials from experiment A as well as all 17 control trials pooled from experiment A and B. Reactions to impossible locations were stronger (Fisher's exact tests: N = 12, p = 0.002; with the controls pooled N = 25, p = 0.005), faster (Fisher's exact test: N = 12, p = 0.018; with the controls pooled N = 25, p = 0.022), lasted longer (Mann-Whitney test: one-tailed, U = 0, z

  reaction, or first head movement occurs more than one second after the emission of the playback 1: First head movement or standing occurs less than one second after the emission of the playback

Figure 1 .

 1 Figure 1. Grunts from (a) an adult female (Gaia, Picnic group) and (b) an adult male (Elvis, Donga group).

Figure 3 .

 3 Figure 3. Results of the group member -stranger experiment. Mean (bold lines); upper and lower quartiles (shaded boxes), minimum and maximum values except outliers (whiskers) and outliers, i.e. more than 1.5 times upper quartile (asterix).

Figure 4 . 27 Figure 5 .

 4275 Figure 4. Results of Experiment A -impossible locations. Upper panels a and b: four controls from Experiment A (AC) only; lower panels c and d: 11 controls from Experiments A and B pooled together (AC, BC1 and BC2). Graph conventions as for Fig. 3.

Table 2 .

 2 Definitions of the variables used in the statistical analyses.

Acknowledgements

We thank the Mpumalanga Parks Board for permission and the University of South Africa (UNISA) for logistic support. Many thanks also to the Applied Behavioural Ecology and Ecosystems Research Unit (ABEERU) colleagues, L. Brown, L. Barrett, S. P.

Henzi, and R. Bshary, who provided essential support. We thank E. Sarroca, C. Scheid, J. Laporte and F. de Castilla for assistance in the field and A. Barrett, C. Fruteau, E. Van de Waal and especially Yan Gobrecht for video-based analysis. L. Barret, R. Byrne, J. Call, J. C. Gómez and K. Zuberbuhler provided helpful comments on earlier drafts. Constructive comments by L. Barrett and an anonymous reviewer helped to improve our text. CNRS ESF-EUROCORES program OMLL (CRP JA33) and NWO program Evolution and Behaviour (051-12-036) provided financial support. Noë & Laporte Socio-spatial Ms.docx


Florence Pérès
email: florence.peres@asconit.com

A Barthès

Etienne Ponton

Michel Coste

Loïc Ten-Hage

René Le-Cohu

Achnanthidium delmontii sp. nov., a new species from French rivers

Keywords: Diatoms, Achnanthidium, Achnanthidium delmontii sp. nov., Languedoc-Roussillon region, France

A new freshwater diatom species belonging to the genus Achnanthidium Kütz. has been found in several French rivers namely those of the Languedoc-Roussillon and Rhin-Meuse regions. Achnanthidium delmontii sp. nov. is described based on light and scanning electron microscopy observations. This species belongs to the group of Achnanthidium taxa with terminal raphe fissures curved to the same side of the valve and it is compared to the most related species: Achnanthidium pfisteri Lange-BertaLot, Achnanthidium pyrenaicum (Husdedt) KoBayasi, Achnanthidium temniskovae Ivanov et ECtor and Achnanthidium zhakovschikovii PotaPova. The assessment of its ecology was made through physicochemical analysis carried out on sites which are part of water quality monitoring networks. Some comments are made on the rapid expansion of Achnanthidium delmontii sp. nov.

Introduction

 [START_REF] Crawford | The Diatoms -Biology & Morphology of the genera. -747[END_REF]provided a renewed interest in diatom systematic with the creation of many new genera and also the reappraisal of older genera such as Achnanthidium Kützing (1844), later more precisely defined by [START_REF] Bukhtiyarova | Four new genera based on Achnanthes (Achnanthidium) together with a re-definition of Achnanthidium[END_REF]. Presently, the Achnanthidium genus includes two groups: a first group with straight terminal raphe endings and a second group with the terminal fissures deflected to the same side of the valve. Following the survey of samples taken from rivers of the Languedoc-Roussillon region in the framework of water quality monitoring, five species with curved raphe fissures were found. One of them, Achnanthidium delmontii, is proposed as a new species. The four others species, Achnanthidium druartii Rimet et Couté, Achnanthidium latecephalum KoBayasi, Achnanthidium pyrenaicum (Husdedt) KoBayasi and Achnanthidium rivulare PotaPova et Ponader, present a high percentage of relative abundance in some rivers of this region. A. delmontii sp. nov. was recorded in several rivers of the river water monitoring networks in France. Till now, in routine inventories which aim to monitor biological quality of rivers, A. delmontii sp. nov. was misidentified under the name A. pyrenaicum, A. pfisteri Lange-BertaLot or non identified to the species level.

This paper presents the morphology, the ecology and the distribution in French rivers of A. delmontii sp. nov., compares it to the most closely related species and give information on its rapid expansion.

Material and Methods

The samples were collected using the French standard NF T 90-354 [START_REF] Afnor | Norme AFNOR NF T[END_REF] in many different rivers of the Languedoc-Roussillon and Rhin-Meuse regions by different public and private organisations in charge of monitoring water quality control networks (ASCONIT Consultants, AQUASCOP, DREAL Languedoc-Roussillon). The diatoms were sampled on natural substrates such as pebbles and stones. The samples were treated by oxidation with hot 40% hydrogen peroxide and hydrochloric acid and rinsed three times with deionised water. For LM observations, cleaned diatoms were mounted in Naphrax. LM and morphometric measurements were performed with a LEICA DM5000B microscope using a 100x oil immersion objective and a LEICA DFC 420 camera. Measurements on Achnanthidium delmontii sp. nov. were taken from 127 individuals. Relative abundance data were based on LM counts of at least 400 valves. For SEM examinations, cleaned diatoms were dried onto glass coverslips attached to aluminium stubs using nail varnish and sputter-coated with a thin layer of platinum. The observations were performed using a JEOL-JSM 6700F and a JEOL JMS 6490. The temporal changes in abundance of A. delmontii sp. nov. was analysed using existing collection slides (from consultancy firm F. PERES, ASCONIT Consultants, AQUASCOP and DREAL Languedoc-Roussillon) dating back to 2005 for some of the stations. Physico-chemical data were provided by DREAL Languedoc-Roussillon and by the water agency Rhin-Meuse (Table 1). Description LM: valves are linear with rounded apices becoming elliptical in small individuals (Figs 1-68). On the raphe valve, the axial area is narrow. The central area is irregular forming generally a rectangular fascia but a shortened stria can be present on one of the margin side. The raphe is filiform, straight, with distinct central pores. Striae are slightly radial.

Results

Achnanthidium delmontii

On the rapheless valve the axial area is acicular. Striae are parallel to slightly radiate at the apices; in most cases, two striae are slightly more spaced apart in the middle part of the valve. SEM: externally, the raphe is straight with droplet-like proximal endings spaced about 0.8 µm apart (Figs 69,70,74). The terminal fissures are deflected to the same side and terminate on the valve face near the valve margin (Figs 69,70,75). Internally, the proximal raphe endings are curved in opposite directions (Figs 71,72,77) and the distal raphe endings terminate in small helictoglossae (Figs 71,72). The central area is generally irregular, forming a more or less rectangular fascia but can present on one side a shortened stria consisting of one to three areolae (Figs 74,77). On the raphe valve, the striae are composed of 5 to 7 (45 to 57 in 10 µm) more or less rounded areolae (Figs 69,70). Sometimes, the Voigt fault is marked by a shortened stria (Figs 70,72). On the rapheless valve, the striae consist of 6 to 7 areolae (55 à 61 in 10 µm) and two of them are more distant in the middle part of the valve (Figs 73,76).

On both valves, the areolae are internally occluded by hymenes perforated by delicate slits (Figs 79-82). There is a row of elongated areolae on the mantle (Figs 75,76,78) with a gap at the poles; the mantle elongated areolae are separated from those of the valve face by a hyaline area (mg.l -1 ) 6.3-7.3 5.7-6.9

5.4-7.5 along the edge of the valve face (Figs 75,76).

Ecology, associated diatom flora and distribution of Achnanthidium delmontii sp. nov.

The type population of Achnanthidium delmontii sp. nov., was found in the river Cèze at Chusclan (Gard, Languedoc-Roussillon). It is a Mediterranean river (20 to 50 m wide), which takes its source in the mountain Cévennes and emerges in the downstream part of the Rhône. The type population of A. delmontii has its maximum relative abundance in a location characterized by an alkaline pH, high conductivity and calcium concentration (Table 1). The PO 4 -P -level in the river Cèze at Chusclan (maximum PO 4 -P - concentration: 75 µg.l -1 ) indicates an eutrophic environment (Table 1). These values are quite close to those observed for A. latecephalum and A. druartii (see also [START_REF] Couté | Achnanthidium druartii sp. nov. (Achnanthales, Bacillariophyta): A new species invading European rivers[END_REF] in the Languedoc-Roussillon region. When A. delmontii was strongly dominant (75%) in the river Cèze at Chusclan, it was associated with Amphora pediculus (Kützing) GrunoW (4%), Achnanthidium minutissimum (Kützing) CzarneCki (4%), Navicula cryptotenella Lange-BertaLot (3%) and Gomphonema minutum Agardh (2%). The associated taxa are mainly β-mesosaprobe and eutrophic, according to [START_REF] Van Dam | A coded checklist and ecological indicator values of freshwater diatoms from the Netherlands[END_REF], which tend to corroborate the chemical data (Table 1).

A. delmontii appears for the first time in 2007 (Fig. 85) with a low percentage of relative abundance (<1%) in one sample: Vieux Rhin in Kembs (VR). In the type location (CZ), A. delmontii appears from 2008 and after that, proliferates (>60%). At the same period, A. delmontii is observed for the first time in rivers such as Canal de Huningue (CH), Canal de Neuf-Brisach (CN), Canal de la Hardt (CH), Ehn (EH), Grand Canal d'Alsace (GC), Ill (IL1 and IL2), Lertzbach (LB), Muhlbach de la Hardt (MH), Quatelbach (QB), Rhin (RH1, RH2, RH3), Vieux Rhin (VR), Rhône (RO1 and RO2) and Cèze (CZ) (Fig. 85). The rapid expansion of A. delmontii seems to correspond to the criteria defined by [START_REF] Coste | Diatomées invasives exotiques ou rares en France: principales observations effectuées au cours des dernières décennies[END_REF] which qualify invasive species (recent appearance, proliferation, rapid expansion).

Discussion

In LM, Achnanthidium delmontii sp. nov. most closely resembles A. pfisteri Lange-BertaLot [START_REF] Werum | Diatoms in springs from Central Europe and elsewhere under the influence of hydrogeology and anthropogenic impacts[END_REF]. A. delmontii is much longer than A. pfisteri (7.3-21.4 µm vs 7-12 µm, Table 2) and has a higher number of striae on the raphe valve (20-26 vs 18-20 in 10 µm, Table 2). Moreover, in A. delmontii, the raphe valve has a well delimited fascia in most cases whereas in A. pfisteri, the central area is asymmetrical (Werum & Lange-BertaLot 2004, plate 20, figs 12-13). In SEM, the terminal fissures, deflected to the same side, are much shorter in A. pfisteri than in A. delmontii. On the raphe valve, the number of areolae per stria is higher in A. delmontii (Table 2). The mantle areolae row is interrupted at the apices on the rapheless valve in A. delmontii whereas it is continuous in A. pfisteri (Werum & Lange-BertaLot 2004, plate 20, fig. 14).

In LM, the presence of a fascia on the raphe valve is a feature shared by A. delmontii and A. temniskovae Ivanov et ECtor (Ivanov & ECtor 2006, figs 1-11). Otherwise, both species can be distinguished by the valve morphology and the dimensions (Table 2). In SEM, the raphe valve striae are subparallel towards the apices in A. delmontii whereas they are strongly radiate in A. temniskovae (Ivanov & ECtor 2006, figs 27, 28). The number of areolae per stria is very different in both species and, moreover, the row of mantle areolae does not stop at the apices in A. temniskovae, unlike A. delmontii.

In LM, A. delmontii and A. zhakovschikovii PotaPova (PotaPova 2006) have linear to elliptic valves with rounded apices. Both species differ in In LM, A. delmontii can be differentiated from A. pyrenaicum by the valve morphology (KoBayasi 1997, figs 3-6;PotaPova & Ponader 2004, fig. 105). A. delmontii is characterized by a fascia on the raphe valve (Figs 1-16) whereas the central area is asymmetrical with often a stria on one side in A. pyrenaicum (KoBayasi 1997, fig. 11;PotaPova & Ponader 2004, fig. 105). Length being equal, A. delmontii is wider than A. pyrenaicum (personal observation). In SEM, in external view, the areolae are rounded on both valves in A. delmontii whereas they are transapically elongated in A. pyrenaicum (Fig. 83, KoBayasi 1997, fig. 15;PotaPova & Ponader 2004, figs 105, 106). On both valves, A. delmontii has 5 to 7 areolae per stria against only 3 to 4 in A. pyrenaicum (Table 2). On the rapheless valve, the row of mantle striae is interrupted at the apices in A. delmontii (Fig. 78) whereas it is continuous in A. pyrenaicum (Fig. 83).

To conclude, we draw attention to the detailed ultrastructure of the areolae. A. delmontii (Fig. 82), A. latecephalum (Fig. 84), A. convergens (KoBayasi) KoBayasi (KoBayasi 1997, fig. 68) and A. japonicum (KoBayasi) KoBayasi (KoBayasi et al. 2006, pl. 155, figs 17, 18) have areolae internally occluded by hymenes perforated by delicate slits. A few Achnanthidium species with straight raphe such as A. saprophilum (KoBayasi et Mayama) round et Bukhtiyarova (Lange-BertaLot & krammer 1989, plate 56, fig. 8;KoBayasi et al. 2006, pl. 161, fig. 17) and A. pusillum (GrunoW) CzarneCki (KoBayasi et al. 2006, pl. 158, fig. 14) show areolae with a cribrum. This feature is possibly another criterion to separate both Achnanthidium groups. Nevertheless, further investigation is required.

Aknowledgements

Material examined from Languedoc-Roussillon region was obtained in the framework of the research programme « Atlas des Diatomées de Languedoc-Roussillon ». Financial support for this research programme was provided by DREAL Languedoc-Roussillon. We gratefully acknowledge C. Deshayes and S. Le Blond-Duplouy for technical assistance with SEM, Agnès Pierret and Florence Barange for the illustrations, Catherine Vignon for the English translation and Aquascop for providing biological material.

Figs 1 -

 1 Figs 1-68. Achnanthidium delmontii sp. nov. LM: (1-16) raphe valves; (17) girdle view; (18-33) and rapheless valves from type population (river Cèze at Chusclan); (34-41) rapheless valves; (42-48) raphe valves; (49) and girdle view (river Rhône at Roquemaure); (50 -54) rapheless valves; (55-63) raphe valves; (64-66) girdle view; (67-68) and abnormal forms (67-68) (river Vieux Rhin at Kembs). Scale bar 10 µm.

  -0.077 0.026-0.068 0.032-0.042 0.032-0.039 0.032-0.039 0.032-0.202 0.033-0.075 0.033-0.042 0.014-0.034 0.033-0Oxygen Demand; Chl a = Chlorophyll a; COD = Chemical Oxygen Demand; Cond = Conductivity at 25°C; DO = Dissolved Oxygen; DOC = Dissolved Organic Carbon; MeS = Suspended Matter; Pheo = Pheopigments; TP = Total Phosphorus; Turb = Turbidity. Table 2. Comparison of morphological characteristics of 5 Achnanthidium morphologically close.* striae in the middle part of the valve; NC = Not communicated; L -length,

Figs 69- 82 .

 82 Figs 69-82. Achnanthidium delmontii sp. nov. SEM: (69) external view of the raphe valve; (70) external view of the raphe valve showing a voigt fault (arrowhead); (71) internal view of the raphe valve; (72) internal view of the raphe valve showing a voigt fault (arrowhead); (73) internal view of the rapheless valve; (74) external view of central area with shortened stria composed of one areola; (75) external view of raphe valve showing the terminal fissures and the interruption of the striae at the end; (76) external view of the rapheless valve showing the hyaline area; (77) internal view of central area showing the raphe proximal endings deflected in opposite side; (78) internal view of the rapheless valve showing the interruption of the striae at the end; (79) internal view of the rapheless valve showing the areolae structure; (80) internal view of the raphe valve showing the areolae structure; (81) internal view of the raphe valve areolae; (82) internal view of the rapheless valve areolae. Scale bars 1 µm (69-81), 100 nm (82).

Fig. 83 .

 83 Fig. 83. External view of rapheless valve of Achnanthidium pyrenaicum, SEM, Rebenty river, France. Scale bar 1 µm. Fig. 84. Internal view of the rapheless valve areolae of Achnanthidium latecephalum, SEM, river Hérault, France. Scale bar 100 nm.

Fig. 85 .

 85 Fig. 85. Distribution and temporal changes of Achnanthidium delmontii sp. nov. in percentage of relative abundance from 2007 to 2010.

Pérès, Le Cohu et Barthès sp. nov.

	Diagnosis
	In microscopio photonico: valvae lineari-ellipticae
	apicibus non protractis late rotundatis, lineares quoad
	specimina maxima sed ellipticae quoad specimina
	minima. Longitudo: 7.3-21.4 µm. Latitudo: 3.3-5.1
	µm. Longitudo/Latitudo ratio: 2.0-4.7. Raphovalva:
	area axialis angusta lanceolata; area centralis circiter
	rectangulata margines valvae attingens, interdum una
	curta stria ab altero latere. Raphe filiformis recta cum
	poris centralibus distinctis. Striae subparallelae ad
	parallelas, 20-

26 in 10 µm usque ad 35 in 10 µm ad apices. Araphovalva: valvae plerumque ellipticae. Area axialis acicularis. Striae subparallelae ad parallelas, 18-22 in 10µm usque 25 in 10 µm ad apices; plerumque duae striae remotiores in media parte.

	Holotype:	PC0142667,	Laboratoire	de
	Cryptogamie, Muséum d'Histoire Naturelle.
	Paris, France.
	Isotype: BR-4247, Departement de Bryophyta,
	Thallophyta, Jardin Botanique National de
	Belgique, Meise, Belgique.
	Type locality: Chusclan -Rivière Cèze -France
	(alt : 34 m ; longitude : 1 907 854; latitude:
	787811,198 -Lambert II)
	Etymology: the specific epithet refers to
	Dominique Delmont, diatomist at DREAL
	Languedoc-Roussillon.
	In microscopio electronico: raphovalva leviter
	concava. Rami raphis cum centralibus extremis forma
	guttulae in externo visu et curvatis ad opposita latera
	in interno visu. Fissurae terminales raphis curvatae ad
	idem latus valvae. Area centralis formans fasciam sed
	interdum una stria composita una ad tres areolas ad
	alteram marginem frontis valvae. Striae compositae
	5-7 quadrangularibus areolis (45-57 in 10 µm) in
	valvae fronte et una elongata areola in valvae limbo.
	Araphovalva: area axialis cum angusto sterno leviter
	dilatata in media parte. Striae compositae 6-7 areolis
	(55-61 in 10 µm) in fronte valvae et una elongata
	areola in valvae limbo.
	In duabus valvis: in interno visu, areolarum hymenes
	ex parte perforati poris tenuiter elongatis; striae
	interruptae sunt duabus polis in limbo et hyalina aream
	circum frontem valvae adest.

Table 1

 1 Cont.


E Darles

B Crespin

D Ghazanfarpour

Accelerating and enhancing rendering of realistic ocean scenes

Keywords: Physically based modeling, ocean scenes, graphics data structures, raymarching, sphere tracing

INTRODUCTION

Realistic ocean scenes are more and more present in computergenerated films, motion pictures and electronic games, but creating such scenes is a difficult task because a lot of modeling and rendering aspects need to be addressed. Furthermore, as the ocean surface is usually very large, raytracing is computationally intensive. The most important aspect in this context is the surface model used to represent realistic waves; it should allow realistic waves shape and motion at the lowest possible cost. Remaining issues are lightwater interactions and complex phenomena occurring at the surface, which usually depend on atmospheric conditions: foam, appearing through advection and turbulent diffusion phenomena during the simulation, modifies radiance at the surface sprays, i.e. small droplets created by foam interacting with wind, involve attenuation and diffusion of light above the surface secondorder scattering: the attenuation and diffusion of light within the water body, due to particles in suspension Taking these phenomena in consideration greatly enhances the realism of ocean scenes as long as they rely on physical parameters and are approximated with sufficient accuracy. In a raytracing environment we also wish to reduce computations and memory requirements. This paper addresses all these issues, by proposing a new method based on a set of spheres that combines an efficient surface model and physicallybased complex phenomena, integrated into a unique data structure. After reviewing different surface models and complex phenomena proposed in the literature in section 2, we choose a parametric model based on real measurements that gives realistic results and present an efficient method for computing ray surface intersections based on sphere tracing in section 3. This approach is then extended to include complex phenomena as described above, using the same simple and efficient data structure (section 4). Finally, results are presented in section 5 along with considerations about implementation and performances of our approach.

RELATED WORK

Modeling and rendering realistic ocean scenes has been investigated by many researchers in the past few Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. years. We focus mainly on papers dealing with scenes depicting the ocean far from the shore, i.e. where breaking waves do not appear. Realtime methods are also briefly reviewed although our work is more oriented towards raytracing. A more general stateoftheart report can be found in [START_REF] Iglesias | Computer graphics for water modeling and rendering: a survey[END_REF].

Ocean surface models

Models that attempt to represent the ocean surface accurately can usually be classified in two different approaches: parametric or spectral. Parametric approaches [FR86, Pea86, TB87, CGG01] represent the ocean surface as a sum of periodic functions which describe waves as a circular motion of water particles. This procedural model is very efficient but does not yield realistic results. On the other hand, spectral methods [MWM87,Tes01] are based on oceanographic measures, synthesized by spectral analysis and hence represent the ocean surface as a height field computed from a sum of sinusoids of various amplitudes and phases; smallscale waves and ripples are modeled directly by adding noise perturbation. This approach ensures high realism, but is not easily controllable. To overcome these problems, hybrid, procedural models were proposed [START_REF] Premoze | Rendering natural waters, Computer Graph[END_REF][START_REF] Thon | Ocean waves synthesis and animation using real world information[END_REF]. The parameters can be obtained automatically from real oceanographic spectra, which allows realistic animations of the ocean surface. Another advantage of the model is its implicit definition, which is well adapted to raytracing. Realtime representations of hybrid models can also be obtained by generating an optimized surface mesh only where necessary, thus reducing both the sampling and the number of waves on the fly [START_REF] Hinsinger | Interactive animation of ocean waves[END_REF][START_REF] Mitchell | Realtime synthesis and rendering of ocean water[END_REF][START_REF] Hu | Realistic realtime rendering of ocean waves[END_REF]. But realtime methods can hardly handle complex phenomena reviewed in the next section.

Lightwater interactions and complex phenomena

The model proposed in [START_REF] Premoze | Rendering natural waters, Computer Graph[END_REF] includes the apparition of foam on waves crests, which locally modifies the optical properties of the ocean. The surface is discretized and rendered by a Monte Carlo path tracer, which also takes into account the turbidity of the ocean, i.e. the opacity of the surface depending on particles suspended in the water. This parameter allows to determine the amount of light scattered towards the surface, and provides realistic simulations of several types of water. Second order scattering is also taken into account in [START_REF] Iwasaki | A volume rendering approach for sea surfaces taking into account second order scattering using scattering maps[END_REF] by computing radiance at sampled points on a set of horizontal slices within the water volume using volume rendering graphics hardware.

Complex phenomena, such as foam, spray or splashes are usually modeled and rendered using particle systems [START_REF] Jensen | Deepwater animation and rendering[END_REF][START_REF] Takahashi | Realistic animation of fluid with splash and foam[END_REF][START_REF] Holmberg | Efficient modeling and rendering of turbulence water over natural terrain[END_REF]. In [START_REF] Thürey | Animation of open water phenomena[END_REF], animation of drops and smallscale interactions such as ripples or splashes are obtained by combining a twodimensional water surface simulation and a threedimensional fluid simulation along with a physically based model. As a conclusion, among these methods only one [START_REF] Premoze | Rendering natural waters, Computer Graph[END_REF] seems able to provide realistic ocean scenes, by combining an efficient surface model and physicallybased complex phenomena. Our work follows the same goal, by integrating these different aspects into a unique data structure. Our implementation uses the ocean surface model of [START_REF] Thon | Ocean waves synthesis and animation using real world information[END_REF], where the reader can find more details. The model is based on a summation of elementary waves called trochoids (see Fig. 1a), which yields realistic waves shapes varying from a smooth profile in the case of calm ocean, to a sharpened crested shape in case of an agitated ocean surface due to greater wind activity. Each trochoid is defined as a displacement function applied to a reference plane, parameterized by its amplitude A i , its frequency f i , its wavelength λ i , its direction of propagation in the plane θ i and its phase ϕ i . Several methods are proposed in [START_REF] Thon | Ocean waves synthesis and animation using real world information[END_REF] to extract correct parameters from real measurements. All the pictures in this paper were obtained using PiersonMoskowitz spectrum; highfrequency trochoids were amplified to represent capillary waves created by the wind. Realistic ocean waves are obtained using 50 different trochoids or more. The counterpart is that a lot of evaluations of trigonometric functions (or access to a precomputed, lookup table storing trochoids) are needed to compute the displacement of a single point; this is particularly critical when rendering ocean scenes using ray marching, since it involves computing many intersections between rays originating from the camera and the ocean surface.

RAYWAVES INTERSECTIONS

Original sphere tracing formulation

Consider the zeroset of an implicit function F: R 3 R → , i.e. the set of points p ∈ R 3 that satisfy F(p) = 0. Bruteforce ray marching attempts to find the intersection between a ray and such a surface using a fixed, constant increment ∆. Let p be a point along the ray such that F(p) > 0, and let p' = p + ∆ be another point along the ray such that F(p') < 0; in that case, the ray crosses the surface somewhere between p and p'. Otherwise the process continues for p' and p''= p' + ∆. The choice of ∆ is critical: if it is too small, the whole process requires a huge amount of evaluations of the implicit function, if it is too large intersections can be missed. A solution to this problem is the sphere tracing method [Har96, HW96], applicable for implicit surfaces that exhibit a Lipschitz bound L such that |F(p)F(q)| ≤ L||pq||, ∀ p,q ∈ R 3 . In that case, the bruteforce algorithm can be enhanced by computing ∆ at each evaluation step by: ∆ = F(p) / L. This guarantees that the next point along the ray p' = p+ ∆ satisfies F(p) ≥ 0, making it possible to converge very quickly to the intersection (see Fig. 1b). The term "sphere" refers to the fact that ∆ represents the radius of a sphere guaranteed not to intersect the zeroset of the implicit surface. Suppose that F is defined as a sum of N functions:

F  p= ∑ i=1 N n i  p 1
Unenhanced sphere tracing in that case would use the global Lipschitz bound L of function F; but this would also require to evaluate each individual term at every step along the ray. It can be noted that the sorting step still seems to require the evaluation n i (p) of each component; actually an approximated value based on the previous iteration is used instead. Experiments using this method show speedups by at least a factor 10 compared to bruteforce raytracing, for fractal surfaces defined by about 10 individual components.

Sphere tracing the ocean surface

In the case of our ocean surface model, and assuming that the equation defining the reference plane is y=0, the displacement n i at a point p:(x, y, z) at time t is defined as:

n i  p = y-A t / N -A i trochoid  k i  x * cos  i z * sin  i - i * t  i  2 
where N is the total number of waves, At is a global amplitude term,  i = 2f i , k i = 2 / λ i and trochoid describes the motion of a water particle. Thus, the implicit function has the same form as in Eq. 1. In order to apply Algorithm 1, we still need to address several issues: choice of a minimum m i for each component: since each component is bounded, its lowest value occurs when (yA t )/N=A i and the trochoid reaches its maximum value. In that case, the global minimum is given by m i =2A i (see Fig. 1a).

choice of a Lipschitz bound L i for each component: in the case of a displacement function applied to a reference plane (such as a trochoid), we can simply use its maximum derivative. For our model, this maximum slope is obtained when the term trochoid(k i (x cosθ i + z sinθ i ) -w i t + ϕ i ) = 1 antialiasing: as in [TG02], a simple antialiasing technique consists in progressively reducing the amplitudes of highfrequency components depending on the projection of the corresponding pixels on the screen (see Fig. 2). In our case, unnecessary components are neglected during the computation of the global efficient bound L e , thus reducing the overall computation cost.

Coherence and data structure

As noted in [START_REF] Hart | Hyperrendering of hypertextured surfaces, Implicit Surfaces[END_REF], spatial coherence can be exploited since the evaluation n i (p) is bounded:

n i  p '-L i ∥p-p'∥n i  p n i  p ' L i ∥p-p '∥
where L i is the Lipschitz bound and p' is a neighboring point. We can now estimate the value of n i (p), provided that n i (p') was evaluated before. This estimation is used to perform the prerequisite sorting stage of Algorithm 1, thus avoiding numerous function evaluations; as a counterpart, the global Lipschitz bound L e may not be a maximum depending on the distance ||pp'||. In our implementation, we use estimates if ||pp'|| is lower than a maximum distance, which is larger than the distance between successive points along the ray. Image coherence can also be exploited directly by the sphere tracing algorithm since, for a given frame, a sphere computed along a ray is guaranteed not to contain an intersection with the surface. Thus, if a ray intersects a sphere computed for a previous ray, then it may progress safely to the next intersection with this sphere. Otherwise, the sphere tracing is performed as usual (see Fig. 3a). It is noticeable that exploiting image coherence requires storing successive spheres along rays. Our data structure is a quadtree, aligned with the reference plane, which stores successive spheres along rays indexed by their intersection with the surface (see Fig. 3b). Any accelerating structure could be employed, but a quadtree seems a natural choice since fluid surfaces usually have regular shapes (especially ocean waves) which adapt well to axisaligned partitioning. Following [START_REF] Hart | Hyperrendering of hypertextured surfaces, Implicit Surfaces[END_REF], image coherence is thus implemented by searching for previously computed spheres in a breadthfirst manner, recursively investigating the neighbors of a ray at the same level in the quadtree. It is also worth noticing, as shown on Fig. 3b, that this data structure is welladapted to antialiasing: subdivision simply depends on the distance to the camera. Our data structure is also useful to exploit temporal coherence between two successive frames if the camera does not move. In this case, spheres stored for a single ray at frame t, as well as the raysurface intersection point p t , can be reused for the same ray at frame t+1 (see Fig. 4):

1. Existing spheres from frame t are marked as old;

2. Sphere tracing is performed backwards (i.e. toward the camera) by recomputing the radius of existing spheres and marking them as new, until the maximal amplitude of the surface is reached. New spheres are also added to fill the gaps that may result; 3. The algorithm stops if the last intersection found is closer to the camera than p t (Fig. 4, left); 4. Otherwise, sphere tracing is performed forwards starting from p t (Fig. 4, right). This strategy can also be adapted for image coherence: if a sphere intersecting a ray was marked as 'old', its radius is simply recomputed and marked as 'new'.

LIGHTWATER INTERACTIONS AND COMPLEX PHENOMENA 4.1 Foam

We are interested in rendering foam which appears in ocean scenes far from the shore, also called passive foam, due to advection and turbulent diffusion of bubbles near the surface under determined conditions (i.e. wind speed greater than 13km/h). Fig. 5 shows a real picture of this phenomenon.

Figure 5. Real ocean scene with passive foam

We do not address in this paper active foam due to breaking waves, which has shorter lifetime and surface coverage, since our surface model does not represent this type of waves. One approach to foam rendering in computer graphics makes use of particle systems [START_REF] Holmberg | Efficient modeling and rendering of turbulence water over natural terrain[END_REF] to simulate life and death of bubbles at the surface. Realistic results are obtained, but this solution can't be applied in the case of large ocean scenes. Another solution is to compute the amount of foam at a point on the surface according to different criteria: height variation between its neighbors [START_REF] Jensen | Deepwater animation and rendering[END_REF], or amplitude of waves propagating around it [START_REF] Jeschke | A procedural model for interactive animation of breaking ocean waves[END_REF], which yields more foam in highfrequency areas. Unfortunately foam does not move in a realistic way between successive frames of an animation. Therefore, as in [PA01], we prefer another approach which considers different atmospheric conditions occurring in the process of foam apparition based on oceanographic results [START_REF] Monahan | Oceanic Witecaps: Their Role in Air Sea Exchanges Processes[END_REF].

Position

The amount of foam s covering the surface at a given point depends on wind speed U 10 measured at 10 meters above the surface, and the difference between water temperature T w and air temperature T a : s=1.59 * 10 -5 * U 10 2.55 * e 0.086 * T w -T a  3 This empirical formulation is used in oceanographic research; but experiments show that it does not take into account the apparition of foam in regions more exposed to wind, especially on waves crests, nor in perturbed regions. Therefore, we propose to extend Eq. 3 in order to include the height of the considered point, and its local slope, thus obtaining the amount of foam s':

s '= s *  y-y min y-y max  n * ∥  N∥ 4
where y min (resp. y max ) is the minimal (resp. maximal) height and  N is the (non normalized) normal vector at the considered point. Parameter n allows the user to control the spread of foam on waves crests; based on empirical experiments, our most visually convincing results were obtained with n ∈ [6, 9]. To avoid noticeable foam patterns repetitions at the surface, a noise function is added, filtered according to the distance to the viewer in the same way as the antialiasing technique described in section 3.2.

Radiance

Here again, an empirical formula for the radiance L foam is suggested by oceanographers [START_REF] Koepke | Effective reflectance of ocean withecaps[END_REF]:

L foam =s ' * f ef * R e where R e is the radiance of pure white foam. The efficiency factor f ef is originally defined as 0.4 + 0.2; to account for the foam's age, we modify it into 0.4 0.2.t.10 2 (truncated to 0 if negative) where t is the time parameter. Total radiance at the surface Ltot is then:

L tot = L 0 1-k r  L r 0
L 0=L sky L sun cos sun  L foam where L(0) is the radiance on the surface computed by taking account the luminance of the sky Lsky (resp. sun Lsun), kr is the Fresnel reflexion coefficient and Lr(0) is the refracted radiance (see section 4.3). To take into account the motion of foam during an animation, we use a texture in image space to store positions and ages of foam on the surface. For each new frame, a preprocessing step consists in incrementing age and displacing positions using formulas similar to Eq. 2. Then, for each pixel, the distances between its projected point on the surface and the neighboring points from the previous foam texture are considered:

1. if the distance is lower than a given threshold, the corresponding texel is used 2. if all the distances are greater than the threshold, a new texel is created in the foam texture using Eq. 4. These distances are computed as soon as an intersection point between the corresponding ray and the ocean surface is found, thus the integration of this computation in our sphere tracing algorithm is straightforward.

Figure 6. Foam obtained with U 10 = 13 km/h Foam obtained with this method is shown on Fig. 6 for U 10 = 13 km/h, T a = 20°C and T w = 12°C.

Sprays

Oceanic spray can be seen as a gaseous exchange between the ocean and the atmosphere. It is mainly due to bubbles bursting at the surface, which release drops of diameter approximately one magnitude lower than bubbles'. Depending on their size, these drops can either go back in the water or evaporate. In the latter case, organic or gaseous molecules carried by the drops are left in the atmosphere, and thus form oceanic sprays. In computer graphics, most existing methods simulate sprays using a particle system generated according to the surface height [JG01, HW04] or surface variations [START_REF] Jeschke | A procedural model for interactive animation of breaking ocean waves[END_REF]. But these methods do not take into account the different atmospheric conditions, and do not simulate the path of light going through sprays (as a participating medium). On the other hand, participating media in computer graphics is the subject of numerous papers (see [START_REF] Perezcazorla | Global illumination techniques for the simulation of participating media[END_REF] for a survey). A method is proposed in [START_REF] Preetham | A practical analytic model for daylight[END_REF] to characterize light attenuation and diffusion due to different atmospheric molecules in daylight. We propose to extend this work to the case of oceanic sprays. Size distribution of sprays of radius r can be empirically computed as [START_REF] Monahan | Oceanic Witecaps: Their Role in Air Sea Exchanges Processes[END_REF]: dF r  dr =1.373U 10 3.41 r -3 10 1.19 10.0057r where s' is the amount of foam computed from Eq. 4 at the orthogonal projection on the surface. Furthermore, the vertical profile of sprays is described by [START_REF] Schettle | Models for the aerosols for the lower atmosphere and the effects of humidity variations on their optical properties[END_REF]:

dF r dr = dF r ,0 dr e -z H max
where dF(r,0) is the size distribution at the ocean surface computed from Eq. 6, and H max is the maximal height where sprays disappear. We now have a complete description of sprays distribution above the surface.

According to BeerLambert's law, the intensity of a light path going through a participating medium is diffused and attenuated. In the case of sprays, this attenuation is due to the sprays' optical thickness (AOD), defined by [PSB99]: During sphere tracing, every time a new step is taken along a ray, the corresponding point is projected onto the surface and the amount of foam is computed, yielding to an attenuation coefficient defined for each sphere below z max (since our algorithm makes use of spatial coherence as described in section 3.3, some spheres can be reused). The diffuse radiance for a given ray is obtained by integration of the diffuse radiance in all M spheres along this ray:

 aod = ∫ 0 z max k e z dz , k e = ∫
L 0=L foam  L sky L sun cos sun  ∑ i =1 M e -k e i  z  z
where ∆z is the vertical distance between two successive spheres.

The influence of sprays on the scene is shown on Fig. 7 for U 10 = 18.5 km/h, T a = 20°C and Tw = 12°C, and is noticeable only because light is attenuated.

Second order scattering

As in the previous section for the atmosphere, light is also greatly attenuated when going through water (a very small amount of light reaches 100m under the surface). This phenomenon is called second order (or subsurface) scattering. Light diffusion in that case is mainly due to particles in suspension in the water volume, which can either be organic We replace the term E d (0) in Eq. 9 by the diffuse radiance L(0) computed by Eq. 7:

L df 0 = 0.33 * b b * L  0/a In that case, the ray is pointing downwards; the bottom of the sea is modeled as a set of trochoids, which allows to apply the sphere tracing algorithm to compute its intersection with a refracted ray. As in the previous section, a diffuse radiance is computed for each sphere by simplifying Eq. 8:

L df z = L df 0 1-e -c-K d z  10
with z the vertical distance between the surface and the center of the sphere. Finally, the total diffuse radiance for a refracted ray is obtained by integrating the diffuse radiance of all M spheres along the ray:

L r 0= Z bottom e -cZ bottom  ∑ i =1 M L df i  z l 11
where ∆l is the distance along the ray and L dfi is the diffuse radiance of each sphere computed by Eq. 10.

In this case again, spatial coherence allows us to reuse diffuse radiance for previously computed spheres. Fig. 8 shows images obtained for clear and turbid waters and constant phytoplancton C = 1.0mg/m 3 .

Implementation

Our method is implemented as a plugin for Autodesk's Maya 6.5. In order to obtain more realistic results, a postprocessing step is added for each frame to include glaring which is particularly frequent in ocean scenes, due to high contrasts between specular neighboring surfaces.

Figure 9. Glaring effect

Our implementation is an extension of a method proposed in [START_REF] Thon | Representation de l'eau en synthèse d'images[END_REF]: the most specular points on the surface are considered, and a "glare texture" is mapped on these points, whose size is determined according to the distance from the viewer. A minimal distance between two specular points is also introduced to avoid a concentration of glare points within a region, which would yield unrealistic results. An example is shown on Fig. 9, where all complex phenomena treated by our algorithm are also enabled (foam, sprays and light scattering).

Performances

Performances of our method were evaluated on different ocean surfaces defined by an increasing number of trochoids waves; the other parameters were constant. As expected, our method guarantees a significant speedup compared to bruteforce ray marching and unenhanced sphere tracing to compute intersections between a ray and the surface (see 1. Speedup compared to bruteforce ray tracing (middle) and unenhanced sphere tracing (right) for 30, 40 and 50 trochoids (N) Spatial coherence was evaluated on a single 640x480 image; temporal coherence was evaluated on a single 640x480 frame of an animation (see videos attached to the paper). Both comparisons were also significant (see Table 2). The total rendering time for a single image listed on the same table shows an interesting result: our method gives better results for a higher number of trochoids. This better convergence is due to our maximization approach, which tends to neglect most of the small amplitude waves with high frequencies. 3 shows the computations repartition between the different parts of the algorithm. Unsurprisingly, most of the time is spent to find intersections between rays and the surface, whereas foam is computed very quickly since most of this part is performed using textures.

N

CONCLUSION

This paper presents an efficient method to render realistic ocean scenes. A spherebased data structure allows to accelerate raysurface intersections computations by heavily relying on spatial and temporal coherence which are welladapted to ocean scenes. To increase the realism of generated scenes, we also propose new formulations to integrate physicallybased phenomena such as second order scattering, foam and sprays into our data structure, without significantly reducing performances. The limitations of our method are mainly due to the surface model, which can not represent breaking waves and physicallybased foam and spray induced by this phenomenon; we are currently studying how to integrate this type of waves.

Figure 1 .

 1 Figure 1. (a) Trochoid example (b) Sphere tracing

Figure 2 .

 2 Figure 2. Simple antialiased scene

Figure 3 .

 3 Figure 3. (a) Image coherence (b) Quadtree structure

Figure 4 .

 4 Figure 4. Starting from spheres traced at frame t (dotted), intersection at frame t+1 can be closer (left) or further (right)

  Figure 7. Light attenuated by sprays

  (phytoplancton) or formed of water molecules of different densities, and characterize the water's turbidity. Different water types (resp. A, B or C) correspond to different turbidities (resp. clear, turbid or very turbid). Due to its complexity, second order scattering can only be simulated using hierarchical data structures: illumination volumes[START_REF] Iwasaki | An efficient method for rendering underwater optical effect using graphics hardware[END_REF], octrees[START_REF] Jensen | A rapid hierarchical rendering technique for translucent material[END_REF] or horizontal planes[START_REF] Iwasaki | A volume rendering approach for sea surfaces taking into account second order scattering using scattering maps[END_REF] were proposed. But these methods do not take particles in suspension into account. Another approach [PA01, CS04] consists in computing the diffuse radiance according to the type of water and its concentration of phytoplancton particles; we propose to simplify and adapt this idea to our data structure. Following[START_REF] Premoze | Rendering natural waters, Computer Graph[END_REF], the directional radiance L(0, , ) θ φ in direction ( , ) θ φ reemitted to the surface is defined as: L 0,  , =L z ,  ,  e -cR L df  zL df  z= L df 01-e -c-K d cos R  8where L(z, , ) θ φ is the radiance at the bottom of the sea, L df (z) is the total diffuse radiance, c is the extinction coefficient of water, K d is the diffusion coefficient, and R = z/(cos ). θ L df (0) is the diffuse radiance at the surface: L df 0= 0.33 * b b * E d 0/ a 9 where b b is the backscattering coefficient, inversely proportional to the phytoplancton concentration C, a is the absorption coefficient, and E d (0) represents the downwelling irradiance at the surface, which depends on atmospheric properties (interested readers should refer to [PA01] for more details).

Figure 8 .

 8 Figure 8. Clear and turbid water 5. RESULTS 5.1 Implementation

Table 1 )

 1 , increasing with the number of trochoids.

	N	W. ray tracing	W. sphere tracing
	30	45%	26%
	40	49%	32%
	50	50%	34%
	Table

Table 2 .

 2 Speedup obtained with spatial coherence (SC) and temporal coherence (TC)

		SC	TC	Time
	30	21%	24%	4:05mn
	40	23%	30%	3:38mn
	50	27%	39%	2:50mn
	Raysurface intersections	62.5%
		Sprays		18.3%
		Second order scattering	12.4%
		Glaring effects		3.7%
		Foam		3.1%

Table 3 .

 3 Repartition of different steps

	Table