Jean-Charles Leclerc
email: leclercjc@gmail.com

Patterns of spatial variability between contrasting substrata: a boulder-field study

Keywords: disturbance, hierarchical design, spatial scale, diversity, intertidal, seaweeds, marine invertebrates, community assembly, NW Brittany, NE Atlantic

Boulder-fields are complex habitats in which many species coexist and are important contributors to coastal biodiversity. These habitats experience important natural disturbances due to wave action, over which anthropogenic stressors, such as hand-fishing and urban sprawl, can be added. Despite their particularity and vulnerability, there have been few attempts to disentangle the processes that actually structure boulder-field communities at different scales.

In order to help direct future research in boulder-field systems, diversity, community structure as well as the patterns of variability were compared between boulder and bedrock habitats at a hierarchy of spatial scales (from 100s cm to 10s km) along approximately 100 km of shoreline in NW Brittany. Specifically, it was hypothesised (i) that difference in physical structure would produce greater variability on boulder than on the bedrock at small scales and (ii) that contrasting patterns of variability will emerge with increasing scales as the processes operating at large scale on the bedrock would be dampened on boulders undergoing physical disturbance.

Overall, both hypotheses were rejected with regards to variability patterns in diversity, community structure and abundances of most functional groups, except for ephemerals in both shore heights and limpets in the low shore. Variability was generally concentrated at the smallest spatial scale, but bedrock showed greater patchiness than boulders for most of the response variables. With increasing spatial scales, the variability patterns were overall consistent between habitats. Among potential mechanisms, the interplay between grazing and physical disturbance over several spatial scales deserve further experimental scrutiny.

INTRODUCTION

Analyzing multi-scale patterns of variability in diversity, abundance and assemblage structure has been proven efficient to highlight the processes likely responsible for observable patterns in population and assemblage structures [START_REF] Underwood | Scales of spatial patterns of distribution of intertidal invertebrates[END_REF][START_REF] Benedetti-Cecchi | Variability in abundance of algae and invertebrates at different spatial scales on rocky sea shores[END_REF]. On rocky shores and more specifically on bedrock substrates, such approaches inferred the significance of small-scale processes, such as biotic interactions (especially competition, facilitation and predation, incl. grazing) and substratum heterogeneity (e.g. [START_REF] Dayton | Competition, disturbance, and community organization: the provision and subsequent utilization of space in a rocky intertidal community[END_REF][START_REF] Mcguinness | Habitat structure and the nature of communities on intertidal boulders[END_REF][START_REF] Guichard | Topographic heterogeneity, hydrodynamics, and benthic community structure: a scale-dependent cascade[END_REF][START_REF] Blanchard | Scales of coastal heterogeneity: influence on intertidal community structure[END_REF][START_REF] Coleman | A continental scale evaluation of the role of limpet grazing on rocky shores[END_REF][START_REF] Smale | Assemblage turnover and taxonomic sufficiency of subtidal macroalgae at multiple spatial scales[END_REF], generally resulting in larger variability at small scales (10s to 100s of centimeters). In contrast to small-scale patterns, the spatial variability at larger scales is generally lower, though processes acting across these scales can result in important differences in population and community structure [START_REF] Broitman | Geographic variation of southeastern Pacific intertidal communities[END_REF], Puente et al. 2017[START_REF] Robuchon | Multi-scale drivers of community diversity and composition across tidal heights: an example on temperate seaweed communities[END_REF]. These patterns can result from variation in coastal topography, wave exposure, upwelling, dispersal and recruitment, generally acting at the mesoscale, i.e. from a few to hundreds of kilometers, as well as climatic gradients, oceanography and geomorphology, which usually act at a broader scale, i.e. from hundreds to thousands of kilometers (see [START_REF] Fraschetti | Patterns of distribution of marine assemblages from rocky shores: evidence of relevant scales of variation[END_REF], Benedetti-Cecchi & Trussell 2014 for reviews). Since all these forces can overlap across spatial scales and interact among each other, identifying a characteristic scale of variation is a robust preliminary approach to determine which processes should be further investigated with thorough experiments. While consistent patterns of spatial variability emerged from general reviews on the topic or cross-system comparisons (e.g. [START_REF] Fraschetti | Patterns of distribution of marine assemblages from rocky shores: evidence of relevant scales of variation[END_REF], Dal Bello et al. 2017), there have been few attempts to compare these patterns between microhabitats shared by rocky shores across several spatial scales. Both the nature and the strength of the forces experienced by these habitats (e.g. bedrock more or less inclined, rock pools, and boulders) may however differ (e.g. [START_REF] Benedetti-Cecchi | The interplay of physical and biological factors in maintaining mid-shore and low-shore assemblages on rocky coasts in the north-west Mediterranean[END_REF][START_REF] Firth | Competition and habitat suitability: small-scale segregation underpins large-scale coexistence of key species on temperate rocky shores[END_REF] and lead to contrasting patterns across spatial scales.

Boulder fields have long been proven to provide a useful framework to study ecological theories (e.g. Sousa 1979a[START_REF] Mcguinness | Species-area relations of communities on intertidal boulders: testing the null hypothesis[END_REF], and yet, rare have been the studies examining multi-scale patterns in these habitats [START_REF] Mckindsey | Body size and spatial variation of community structure in subarctic intertidal boulder fields[END_REF][START_REF] Chapman | Molluscs and echinoderms under boulders: Tests of generality of patterns of occurrence[END_REF]). It has long been suggested that physical disturbance is one of the major forces influencing community structure in boulder fields [START_REF] Osman | The establishment and development of a marine epifaunal community[END_REF], Sousa 1979a[START_REF] Littler | Relationships between macroalgal functional form groups and substrata stability in a subtropical rocky-intertidal system[END_REF]) (but see McGuinness 1987b). By overturning boulders at different rates (turnover), physical disturbance may create a mosaic of patches undergoing different stages of community development.

Though the probability of overturning of individual boulders could be estimated by its size [START_REF] Mcguinness | Species-area relations of communities on intertidal boulders: testing the null hypothesis[END_REF], McGuinness 1987a), it does not necessarily reflect the intensity of disturbance actually operating on such heterogeneous habitats [START_REF] Blanchard | Scales of coastal heterogeneity: influence on intertidal community structure[END_REF]. In addition, the rate of boulder overturning may be locally increased by anthropogenic impacts such as seaweed and seafood harvesting (Le [START_REF] Hir | Macrofaunal diversity and habitat structure in intertidal boulder fields[END_REF][START_REF] Stagnol | Effects of commercial harvesting of intertidal macroalgae on ecosystem biodiversity and functioning[END_REF][START_REF] Hily | Can we use a landscape approach to assess natural and anthropogenic perturbations of the rocky shore ecosystems?[END_REF]. While a huge boulder is unlikely to be overturned by wave action, a smaller boulder may experience an intermediate level of disturbance. When a boulder is overturned for a long period (several months), most of its sessile biota may be killed, and a new sequence of succession is to be started on the open space created upon its top surface (Sousa 1979b).

Depending on its duration, the period of overturning will differently affect the ongoing succession sequence as some biota in place may either persist as reproductive or vegetative stages. By favouring the coexistence of a mosaic of successional stages and preventing dominance, an intermediate level of disturbance can thus be assumed to be associated with a greater species diversity [START_REF] Osman | The establishment and development of a marine epifaunal community[END_REF][START_REF] Connell | Diversity in tropical rain forests and coral reefs. Set Connell SD (2003) Negative effects overpower the positive of kelp to exclude invertebrates from the understorey community[END_REF], Sousa 1979b[START_REF] Sousa | The responses of a community to disturbance: the importance of successional age and species' life histories[END_REF]. Under heterogeneous disturbance, communities on boulder fields may therefore be naturally more variable at the patch scale than on more temporally-constant surrounding bedrock. Further processes interacting with physical disturbance (e.g. grazing) may however make such model more complicated and its outcomes may vary according to life form strategies and specific stress-resistance [START_REF] Sousa | The responses of a community to disturbance: the importance of successional age and species' life histories[END_REF][START_REF] Littler | Relationships between macroalgal functional form groups and substrata stability in a subtropical rocky-intertidal system[END_REF].

The rare studies performed at a hierarchy of spatial scales in boulder fields suggested that less and less variability is explained by increasing spatial scale (e.g. [START_REF] Mckindsey | Body size and spatial variation of community structure in subarctic intertidal boulder fields[END_REF][START_REF] Chapman | Molluscs and echinoderms under boulders: Tests of generality of patterns of occurrence[END_REF][START_REF] Liversage | Disturbance-related patterns in unstable rocky benthic habitats of the north-eastern Baltic coast[END_REF]. Though not jointly performed, similar studies conducted on the bedrock generally captured processes operating at larger scales, such as among shores [START_REF] Benedetti-Cecchi | Variability in abundance of algae and invertebrates at different spatial scales on rocky sea shores[END_REF][START_REF] Valdivia | Variation in community structure across vertical intertidal stress gradients: how does it compare with horizontal variation at different scales?[END_REF] or among regions [START_REF] Bello | Consistent patterns of spatial variability between NE Atlantic and Mediterranean rocky shores[END_REF]. If the disturbances acting upon boulders are sustained across several spatial scales, they may overwhelm processes coming into play with increasing spatial scales (e.g. wave exposure, recruitment) on the bedrock, and contrasting patterns of variability may thus be expected between habitats. Processes operating at larger spatial scales may however interact with these disturbances and the local influences (e.g. species pool) may change the outcomes of this model [START_REF] Chapman | Inconsistency and variation in the development of rocky intertidal algal assemblages[END_REF], for example, boulder communities may tend to converge with bedrock in some place but not in others.

The present study aimed to compare boulder and bedrock substrata across different spatial scales and tidal heights in NW Brittany. In order to make both multivariate and univariate comparisons possible between habitats, bedrock was compared with the top of the boulders. Many fewer exclusive species occur in this stratum compared to habitats underneath boulders, therefore differences in multivariate variability would mainly result from differences in abundance of a common set of species rather than differences in identity. Beyond pattern descriptions, the following hypotheses were tested: first, it was expected that small-scale variability in richness, abundance of most taxa and community structure is lower on the bedrock than on boulders, which experience higher level of disturbance. Second, it was expected that contrasting patterns of variability between habitats will emerge from 10s m to 10s km as processes operating on the bedrock over increasing spatial scales are overwhelmed by the disturbance presumably undergone by boulders at small scale.

METHODS

Sampling design and data collection

The study was performed during November 2015 'Spring' tides along approximately 100 km of shoreline in NW-Brittany (Fig. 1, Appendix S1). In this mega-tidal region, the tidal range may reach 8.5-10 m (increasing along a W-E gradient). This portion of coastline is part of the same well-mixed water body and experiences relatively cold (mean about 12°C) and stable (yearly amplitude of 6-8°C) sea-surface temperatures throughout the year [START_REF] Birrien | Nutrients and primary production in a shallow stratified ecosystem in the Iroise Sea[END_REF][START_REF] Gallon | Twenty years of observed and predicted changes in subtidal red seaweed assemblages along a biogeographical transition zone: inferring potential causes from environmental data[END_REF]. The dominant rock formation is granite, though mixed with micaschist at the westernmost part (locality A, Fig. 1), gneiss at northernmost (B) and gabbro (with pegmatite patches) at easternmost (D). The study region is generally exposed to WNW swell, varying on average from approximately 1.5 m in summer to 3.3 m in winter (with maxima up to 10.5 m), and showing a slight decrease along a W-E gradient (Appendix S). According to the coastline conformation and to the wave energy-gradients, coastal boulder-fields can be encountered either along headlands or embedded within rocky inlets [START_REF] Duff | Les zones Natura 2000 de Corsen et Tréompan Vol 1: Les habitats de l'estran de la zone de Corsen[END_REF].

All boulder-fields (sites) were selected according to the presence of both natural boulders densely clustered and stable bedrock along at least 300 m of shoreline, extending at least from the top fringe of the mid-shore (about. 6 m above chart datum) and the infra-littoral fringe (chart datum ± 1 m approximately). In addition, an operational criterion was adopted: the sampled boulders had to be large enough to be covered by a flexible 0.1 m²-quadrat on their top part and light enough to be overturned manually by the present observer. These individual boulders were thus assumed to experience an intermediate level of disturbance due to wave action (Sousa 1979a) as well as potential disturbance from recreational fishing [START_REF] Hily | Can we use a landscape approach to assess natural and anthropogenic perturbations of the rocky shore ecosystems?[END_REF]. Both habitats (boulders and bedrock) were sampled at two tide-levels (mid-and low-shore), adopting a hierarchical design comprising three random factors (locality, site and patch). Localities were haphazardly selected, 20-30 km apart within the region (Fig. 1). Within each locality, two study sites (the so-called boulder fields) were haphazardly selected 2-5 km apart. Finally, two patches (20-50 m wide) were placed 200-300 m apart within each site. While boulders tend to accumulate along the bedrock according to wave energy-gradients at the scale of a site, it was possible to find -generally on site edges -patches presenting a scattered arrangement of both habitats, hence avoiding confounding effects in habitat comparisons. Within a patch, the wave exposure experienced by these habitats was expected to be the same. Nonetheless, comparisons between patches or tidal heights may involve differences in wave exposure (see Appendix S1 for further details) owing to the site features (headland, island and inlets), and hence contrasting frequency and intensity of disturbance (McGuinness 1987a). Tidal heights were defined using a tide table: 2.0 ± 0.50 m above chart datum for the low-shore and 4.5 ± 0.50 m for the midshore. These heights were considered for both the upper side of boulder and the bedrock (i.e. sampled surface) to avoid confounding effect due to emersion period [START_REF] Mcguinness | Habitat structure and the nature of communities on intertidal boulders[END_REF]). Within each patch and at each tidal height, the stable bedrock (n = 16) and the top of the boulders (n = 16) were sampled using a flexible (plastic mesh) 0.1 m²-quadrat.

Abundances were assessed for sessile organisms (seaweeds, sessile fauna) and mobile invertebrates (> 2 mm) using percentage covers and counts, respectively. Seaweed and sessile fauna percentage cover was assessed by summing over 25 sub-quadrats, within each an abundance score from 1 to 4 was given to each taxon (after [START_REF] Dethier | Visual versus random-point percent cover estimations:'objective'is not always better[END_REF]. In order to take into account species layering, percentage cover was assessed for epiphytic, canopy and understorey species (after moving the canopy aside); therefore the total frequently exceeded 100%. All quadrats were photographed entirely and a series of close-ups (above and beneath canopy) were taken for double-checking in the laboratory. Given the important diversity and layering of low-shore assemblages, their diversity and assemblage structure were directly assessed on the field. As for mid-shore, their assemblages were visually assessed based on photographs later in the laboratory using the same method described above. Most specimens were visually (non-destructively) identified at the lowest taxonomic level possible (generally species). A few specimens were brought to the laboratory, preserved at -20°C, before further identification under a dissecting microscope. Notably, it was operationally unmanageable to remove all limpets, from the genus Patella (a total of 10,866 ind.), from their substrate to allow identification based on foot and tentacle patterns; only the blue-rayed limpet Patella (formerly Helcion) pellucida L. could be identified based on superficial (shell) criteria. P. vulgata L., P. depressa Pennant and P. ulyssiponensis Gmelin were thus pooled into a single limpet group: Patella spp. Given the inefficiency in counting the cryptic periwinkle Melarhaphe neritoides L. from photographs (mid-shore bedrock), this species was not included in analyses.

Statistical analyses

General patterns in species diversity and distributions

Patterns in species richness, community structure (all taxa) and abundance of functional groups of algae (sheet-like, filamentous, coarsely branched, thick leathery canopy, jointed calcareous and crustose, [START_REF] Littler | Relationships between macroalgal functional form groups and substrata stability in a subtropical rocky-intertidal system[END_REF], key faunal taxa (barnacles, limpets) and bare space -shared by both habitats and tidal levels -were examined with a five-way design (Fig. 1) using permutational multivariate (or univariate) analyses of variance (PERMANOVA, [START_REF] Anderson | A new method for non-parametric multivariate analysis of variance[END_REF], with 4999 permutations. Factors were 'habitat' (fixed, 2 levels: boulders and bedrock), 'level' (fixed, 2 levels: mid-and low-shore), 'locality' (random, 4 levels), 'site' (random, 2 levels nested within Locality) and 'patch' (random, 2 levels, nested within Site and Locality). Univariate analyses were based on Euclidian distance matrices whereas multivariate analyses were based on Bray-Curtis similarity matrices generated from either raw or transformed data. In order to down-weight the importance of most abundant species (and homogenize multivariate dispersion), multivariate data were square root-transformed. No transformation allowed homoscedasticity to be achieved in univariate data (PERMDISP analyses, unpresented). Given the balance of the design and the large number of samples (n = 16), univariate PERMANOVAs (analogous to ANOVAs) were considered robust enough to cope with this issue and were run on untransformed data [START_REF] Underwood | Experiments in ecology: their logical design and interpretation using analysis of variance[END_REF]. When appropriate, PERMANOVAs were followed by pairwise comparisons and P-values were estimated using Monte Carlo procedure. In order to examine the general patterns in habitat, vertical and horizontal distribution, a principal coordinate (PCO) analysis and a cluster analysis were jointly performed on centroid values computed from each combinations of Habitat × Level × Patch (Site (Locality)), i.e. from 16 replicates (Appendix S2). Subsequently, additional PCOs were performed on all samples and correlations of the variable to the axes were analysed in order to better explain these patterns. Since multivariate and most univariate PERMANOVAs showed a three-way interaction, separate graphics were presented for mid-and low-shore assemblages, respectively.

Patterns of spatial variability across horizontal scales, habitats and levels

Degrees of variability were independently estimated for all combinations of Habitat (boulder and bedrock) × Level (mid-and low-shore) × horizontal Scale (Patch, Site, Locality and Region). To this end, community structure, richness and abundances of functional groups of algae, barnacles, limpets and bare space in mid-shore boulders, mid-shore bedrock, lowshore boulders and low-shore bedrock were all examined separately using a fully nested threeway PERMANOVA, with Patch nested in Site and Site nested in Locality. For each response variable, these analyses were repeated on eight independent matrices, yielded from the random selection of two quadrats per patch. Untransformed data were used for all multivariate and univariate analyses, respectively calculating Bray-Curtis dissimilarities and Euclidean distances. Pseudo-variance components (or variance in the case univariate analyses) were extracted for each source of variation (within region: 10s km; within locality: 1000s m; within site: 10s m; and within patch: 100s cm) by setting equal the observed mean square to their expectations [START_REF] Anderson | PERMANOVA+ for PRIMER: Guide to Software and Statistical Methods[END_REF]. When occurring, any negative estimate was set to zero and the model was adjusted -by excluding the corresponding factor -to re-calculate the remaining estimates [START_REF] Fletcher | How to cope with negative estimates of components of variance in ecological field studies[END_REF][START_REF] Fraschetti | Patterns of distribution of marine assemblages from rocky shores: evidence of relevant scales of variation[END_REF]. For each response variable, a total of 128 estimates of horizontal variability were thus obtained and examined using a balanced three-way analysis of variance (ANOVA). Factors (all fixed and orthogonal) were Habitat (boulder and bedrock), Level (mid-and low-shore) and Scale (Patch, Site, Locality and Region). Although pseudo-variance estimates were systematically log-transformed to reduce heterogeneity (Benedetti-Cecchi 2001), the homogeneity of variances could not be achieved (Levene's test). As such, a conservative level of significance (α = 0.01) was applied [START_REF] Underwood | Experiments in ecology: their logical design and interpretation using analysis of variance[END_REF][START_REF] Fraschetti | Patterns of distribution of marine assemblages from rocky shores: evidence of relevant scales of variation[END_REF]. Provided that components of variation of residualsassociated with large degree of freedom -are generally more accurately estimated than other terms in nested ANOVA, obtaining heterogeneous variances was not surprising. It should be noted that other procedures herein employed (one-way ANOVAs for each scale, variance or dissimilarity calculations for pairs of samples) led to similar results, and were thus unpresented.

When appropriate, ANOVAs were followed by Student-Newman-Keuls (SNK) tests for multiple comparisons (with α = 0.05). ANOVAs were performed using SigmaPlot, while PERMANOVAs, PCOs and PERMDISPs were performed using PRIMER 7 [START_REF] Clarke | Change in marine communities: an approach to statistical and interpretation[END_REF].

RESULTS

During this study, a total of 114 taxa were identified (2 lichens, 49 seaweeds and 63 animals). Complete lists and species authorities are provided in Appendix S3. Cumulated richness was more than twice higher in the low-shore (104 taxa) than in the mid-shore (43 taxa).

Average abundances (per sites) within the mid-shore and the low-shore are given for both habitats in Appendix S4 and S5, respectively. Across horizontal scales, the preliminary principal component analysis based on centroids suggested differences in assemblage structure among habitats and shore levels, with habitat discrimination seemingly level-dependent (Appendix S2).

Dissimilarities in diversity and assemblage structure between boulders and bedrock

Overall, boulders and bedrock were different, with some inconsistencies across scales and levels depending on the response variable considered (Appendix S6, Fig. 2). According to the PERMANOVAs and pairwise tests (Appendix S6), the more consistent result was regarding richness (interaction Habitat × Level), being about twice as high(average ± SD) on the bedrock than on boulders in the mid-shore (6.2 ± 2.3 vs 3.4 ± 1.4, respectively, Fig. 2A) and in the lowshore (13.2 ± 3.5 vs 7.2 ± 2.7, respectively, Fig. 2A'). Although contrasting between tidal levels, almost consistent results were observed for the abundance of Patella spp. (interaction Habitat × Level, Fig. 2B-B'), which was caused by abundances being similar between habitats in the mid-shore (overall 8.0 ± 6.7) but lower on the bedrock (6.9 ± 8.4) than on boulders (19.1 ± 12.5) in the low-shore. Contrasting and patch-dependent results were observed for the cover of barnacles: covers were lower on boulders than on the bedrock in 11 patches out of 16 in the mid-shore (Appendix S6, Fig. 2C) and greater on boulders than on the bedrock in 3 patches in the low-shore (Fig. 2C'). A three way-interaction (Habitat × Level × Patch (Site (Locality)), appendix S6) was observed for the bare space cover, but pairwise tests revealed relatively consistent results: bare space was lower on the bedrock than on boulders within 13 patches out of 16 in the mid-shore (on average 74.6 ± 24.4 vs 91.0 ± 14.3, Fig. 2D) and within 15 patches in the low-shore (9.8 ± 16.0 vs 54.4 ± 31.0, Fig. 2D'). In contrast, despite a three way-interaction (Habitat × Level × Site (Locality)), appendix S6), no significant differences could be observed regarding the cover of sheet-like seaweeds (here represented by ephemerals, such as Ulva and Porphyra spp., Appendix S4) between habitats in the mid-and low-shore at any site. This group seemed however to be more represented, though not very abundant and highly variable (up to 25.1 ± 24.2 % cover), on boulders than on the bedrock (Fig. 2E-E'). The percentage cover of filamentous algae tended to be greater on the rock than on boulders (Habitat × Level × Patch) in the low-shore (13 patches out of 16, Fig. 2F'), but less clearly in the mid-shore (4 patches, Fig. 2F). Similarly, coarsely branched algae displayed greater cover on the bedrock than on boulders (Habitat × Level × Patch) within 3 patches in the mid-shore (Fig. 2G) and within 14 patches in the low-shore (Fig. 2G'). Less abundant than other groups, jointed calcareous displayed greater cover on the bedrock than on boulders (Habitat × Level × Patch) in only 1 patch in the mid-shore and 8 in the low-shore (Fig. 2I-I'). No difference could be detected regarding the crustose group between habitats (Fig. 2J-J'). In contrast, canopy cover (i.e. thickleathery seaweeds, Fig. 2H-H') was greater on the bedrock than on boulders (Habitat × Level × Patch) within 12 patches out of 16 in the mid-shore (12.0 ± 19.4% vs 2.2 ± 5.6%) and within 13 patches in the low-shore (36.7 ± 31.1% vs 3.8 ± 10.7%).

As for the community structure, differences between habitats appeared patch-dependent (threeway interaction, Appendix S6, Fig. 3), although pairwise tests indicated significant differences between habitats in all patches, either in the mid-and in the low-shore (Appendix S6). In the mid-shore, habitat appeared mainly discriminated by the first axis of PCO (30.9% of variation, Fig. 3A-B), highly correlated with the cover of bare space (r = 0.8), tending to be greater on boulders than on the bedrock. In the low-shore, samples appeared more scattered on the PCO and, in spite of some overlap, habitats were mostly discriminated by the first axis (41.0% of variation, Fig. 3C). Boulders tended to display greater cover of bare space and abundances of Patella spp. in comparison to the stable bedrock, characterized by greater canopy (especially F. serratus) cover and diverse associated species (Fig. 3D).

Horizontal patterns in diversity and assemblage structure within habitats

Important similarities in community structure were observed all along the studied area and no clear patterns in richness and abundances could be identified among localities (Fig. 2, Appendix S6). Some subtle differences could however be observed on the stable bedrock, but not on boulders. For instance, a three-way interaction Habitat × Level × Locality was revealed for community structure and discriminated the western locality "A" from the others on the stable bedrock at both shore heights, and the eastern locality "D" from the others in the midshore (Appendix S6). These differences are well illustrated on the PCOs, though explaining a limited amount of variation (Fig 3), as marked differences are also observed within sites (i.e. among patches, Appendix S6). In the mid-shore, the second axis (23.9%) tends to depict a balance between barnacles/limpets (eventually mussels) and fucoids, respectively more abundant in localities A/D and B/C (Fig. 4A-B, Appendix S4). Further variation (14.7 % on Axis 3, non-presented) discriminated patches partially covered by Fucus vesiculosus var. linearis to patches sheltering some Ascophyllum nodosum and accompanying species. Finally, dissimilarities among localities were also suggested from community structure of the stable bedrock in the low shore. The second PCO axis (9.8% of variation, Fig. 3C-D) tended to differentiate sites with F. serratus canopy to sites dominated by diverse red turf-forming seaweeds (e.g. Osmundea pinnatifida, Chondracanthus acicularis) and alternative canopy (e.g.

Himanthalia elongata) (especially in locality A).

As for pooled morpho-functional groups, horizontal patterns were less evident from univariate analyses and the variability between patches was important within most of the sites, with contrasting patterns depending on the habitat as well as on the shore-height considered (Fig. 2, Appendix S6). Habitat × Level × Locality interactions were observed for the abundances of coarsely branched and crustose seaweeds as well as barnacles, but pairwise tests solely revealed logical groupings for the latter. Locality-to-locality differences in barnacle covers (fig. 2C-C

Patterns of spatial variability across horizontal scales, habitats and levels

Patterns of horizontal variability across scales appeared relatively consistent between habitats and shore levels -generally higher at the scale of patch -although depending on the response variable considered (Table 1-2, Fig. 4). While multivariate components of variability (community structure) displayed greater values on the bedrock than on boulders, it was consistently higher at the scale of the patch. Variability was equally distributed between site and locality, though it increased within the region. While these patterns seem to be driven by values observed on the bedrock (Fig. 4), no interaction (e.g. Scale × Habitat × Level) was significant at the level adopted (α < 0.01, Table 1). Such interactions were significant for the abundance of Patella spp., the covers of bare space, barnacles and jointed calcareous algae (Table 1-2). Nonetheless, the only consistent logical grouping was greater variance values at the scale of patch at both shore levels, with the exception of bare surface in mid-shore bedrock.

In only a few cases, within site-variability appeared higher than within locality, namely for jointed calcareous and crustose algae in the low-shore bedrock, and barnacles in the mid-shore.

Patterns of increasing variability between the scales of locality and region -revealed with overall community structure -were also observed with the cover of barnacles in the mid-and the low-shore and with the cover of crustose algae in the low-shore.

Finally, greater values of horizontal variability were generally observed on the bedrock as compared to boulders (Fig. 4, Table 1-2), most likely due to difference in mean values (e.g. richness and abundance of most functional groups of seaweeds, Fig. 2). Greater variability on boulders as compared to bedrock were only revealed with the cover of sheet-like ephemeral algae within the scales of patch, site and locality, along with the cover of bare surface at the scale of locality in the mid-shore and limpet abundances in the low-shore (Fig. 4, Table 3).

DISCUSSION

Across the majority of sampling sites, boulders appeared to be devoid of most functional groups of seaweeds, found in abundance of the stable bedrock. In addition, the variability in abundance of most of these groups was lower on boulders than on the bedrock irrespective of the horizontal and vertical scales considered, except for jointed calcareous and sheet-like seaweeds. The latter group, dominated by ephemerals actually experienced larger variability in abundance on boulders than on the bedrock across three horizontal scales (from metres to kilometres). While these results could suggest that most replicate boulders were more disturbed than the surrounding bedrock, these differences did not translate to greater patchiness in algal abundances on boulders than on the bedrock. The opposite pattern was actually observed for all groups, except early successional species at both tidal heights and limpets in the low shore.

First, this suggests that boulder patches do not necessarily represent a mosaic of successional stages, but could rather be dominated by individual boulders at an early stage of development.

Second, this indicates that small-scale variability is pervasive within the studied communities, including on the comparatively 'stable' bedrock. While physical disturbance may contribute to the observed patterns, further processes (e.g. grazing pressure, habitat selection) are likely to operate at both tidal heights studied.

Small-scale patterns and variability

At both tidal heights, important variability occurred at the scale of the patch and was generally greater than any other spatial scale considered. This finding aligns with reviews on the topic (e.g. [START_REF] Fraschetti | Patterns of distribution of marine assemblages from rocky shores: evidence of relevant scales of variation[END_REF], regardless of the substrate type (boulder versus bedrock).

Contrary to expectations, patchiness was however generally greater on the bedrock than on boulders, except for ephemerals in both shore heights and limpets in the low shore. In parallel, boulders presented lower covers of barnacles, leathery fucoids and associated species -but higher densities of limpets in the low-shore -and were comparatively less biodiverse at the patch scale. This result may thus partially be explained by mean-variance relationship [START_REF] Taylor | Aggregation, variance and the mean[END_REF][START_REF] Benedetti-Cecchi | The importance of the variance around the mean effect size of ecological processes[END_REF], although further mechanisms could be invoked.

Small scale variability is a common feature of rocky shore communities from moderately to highly exposed conditions and can be influenced by various interplaying processes such as substratum heterogeneity, species behaviour and biotic interactions [START_REF] Dayton | Competition, disturbance, and community organization: the provision and subsequent utilization of space in a rocky intertidal community[END_REF][START_REF] Connell | Community interactions on marine rocky intertidal shores[END_REF][START_REF] Raffaelli | Intertidal Ecology[END_REF][START_REF] Underwood | Scales of spatial patterns of distribution of intertidal invertebrates[END_REF][START_REF] Guichard | Topographic heterogeneity, hydrodynamics, and benthic community structure: a scale-dependent cascade[END_REF][START_REF] Raimondi | Processes structuring communities: evidence for trait-mediated indirect effects through induced polymorphisms[END_REF][START_REF] Fraschetti | Patterns of distribution of marine assemblages from rocky shores: evidence of relevant scales of variation[END_REF]. NW European mid-shores are generally characterized by clumps of either Ascophyllum, Fucus or Mytilus (all observed in the present study), whose formation depends on hydrodynamic forces acting at the scale of hundreds of meters and biotic interactions at the patch scale. For instance, in moderately exposed conditions, Fucus vesiculosus may establish within dense clusters of barnacles, which provide substratum and refuge from grazing by limpets. Adult plants may reduce further settlement of barnacles through sweeping while favouring aggregations of limpets and barnacle-predators under the canopy, and in turn limit additional plants and barnacles to establish at the vicinity [START_REF] Hawkins | Grazing of intertidal algae by marine invertebrates[END_REF][START_REF] Hartnoll | Patchiness and fluctuations on moderately exposed rocky shores[END_REF]. Through habitat-formation [START_REF] Jenkins | Direct and indirect effects of a macroalgal canopy and limpet grazing in structuring a sheltered inter-tidal community[END_REF][START_REF] Moore | Role of biological habitat amelioration in altering the relative responses of congeneric species to climate change[END_REF], both richness and species interactions are promoted at the patch scale, hence increasing the small scale variability in richness and overall community structure on the bedrock. In the low shore, diverse assemblages of seaweeds and associated invertebrates were observed. The abundance of canopy species varied among shores and some differences in community structure were detected on the bedrock.

Nonetheless, there was no dominance of a single canopy species (thick leathery cover varied from 4 to 76% cover per patch), nor exclusion of understorey seaweeds. The variability in the abundance of thick leathery groups was greater at the scale of the patch and this pattern was tracked by all seaweed functional groups, except sheet-like ephemerals. While dominant and monospecific canopy stands could eventually exclude most understorey species owing to shading, whiplash and scouring [START_REF] Wernberg | Modification of the physical environment by an Ecklonia radiata (Laminariales) canopy and implications for associated foliose algae[END_REF], such negative interactions may be dampened as environmental factors, such as the substratum heterogeneity (topography, sediments) and hydrodynamics, come into play (Connell 2003[START_REF] Toohey | Canopy-understorey relationships are mediated by reef topography in Ecklonia radiata kelp beds[END_REF]. A patchily distributed and/or multi-specific canopy would be more likely to create a mosaic of environmental conditions (light gaps, protection from physical stress) and to promote diverse assemblages within which multiple interactions (either direct or indirect; positive, neutral and negative) take place [START_REF] Benedetti-Cecchi | Testing the relative contribution of positive and negative interactions in rocky intertidal communities[END_REF][START_REF] Irving | Physical disturbance by kelp abrades erect algae from the understorey[END_REF][START_REF] Smale | Subtidal macroalgal richness, diversity and turnover, at multiple spatial scales, along the southwestern Australian coastline[END_REF][START_REF] Bulleri | Temporal stability of European rocky shore assemblages: variation across a latitudinal gradient and the role of habitat-formers[END_REF]. Like in the mid-shore, the greater patchiness observed on the bedrock as compared to boulders could thus simply be explained by greater biological interactions as well as higher species diversity and abundances of almost all successional groups of seaweeds. In addition, the substratum complexity and the presence of dense boulders surrounding the bedrock may also affect the incident light and hydrodynamics (current velocity and turbulence) from centimetres to meters [START_REF] Guichard | Topographic heterogeneity, hydrodynamics, and benthic community structure: a scale-dependent cascade[END_REF][START_REF] Guichard | Scaling the influence of topographic heterogeneity on intertidal benthic communities: alternate trajectories mediated by hydrodynamics and shading[END_REF][START_REF] Mckindsey | Body size and spatial variation of community structure in subarctic intertidal boulder fields[END_REF]. Even more likely, given the close proximity of bedrock and boulders at the scale of patch, disturbance may also contribute to small-scale variability on the bedrock [START_REF] Shanks | Adding teeth to wave action: the destructive effects of waveborne rocks on intertidal organisms[END_REF][START_REF] Povey | Effects of trampling on plant and animal populations on rocky shores[END_REF]. Bare space may be opened through boulders hitting bedrock, sand scouring and recreational fishing (seaweed and shellfish removal, stamping), hence providing settlement substrate and refuge from grazers to various seaweeds, regardless of their expected successional stages [START_REF] Sousa | The responses of a community to disturbance: the importance of successional age and species' life histories[END_REF].

Seminal works on boulder fields supported the intermediate disturbance hypothesis (e.g. [START_REF] Osman | The establishment and development of a marine epifaunal community[END_REF], Sousa 1979a) as greater diversity and patchiness (mosaic of successional stages) were observed on moderately disturbed boulders, while stable (or stabilized) boulders tended to be dominated by competitive dominants, such as Gigartina sp. (Sousa 1979a[START_REF] Sousa | The responses of a community to disturbance: the importance of successional age and species' life histories[END_REF] excluding other species through competition for resources (light, substratum) and whiplash securing the open space. However, this model does not hold for all boulder fields, systems within which physical disturbance is unlikely to explain alone community dynamics [START_REF] Mcguinness | Habitat structure and the nature of communities on intertidal boulders[END_REF], McGuinness 1987a, b). In the mid-shore, limpet abundances were statistically similar between habitats (Appendix S6). Since boulder overturning has not been thoroughly inferred in the present study, it could be hypothesised that physical disturbance is locally unimportant [START_REF] Bishop | Imbricate and fitted fabrics in coastal boulder deposits on the Australian east coast[END_REF] or -alternatively -that limpets are highly resistant to such disturbance, perhaps due to their anchoring capacity and mobility [START_REF] Raffaelli | Intertidal Ecology[END_REF] and to boulder substratum heterogeneity [START_REF] Mcguinness | Habitat structure and the nature of communities on intertidal boulders[END_REF].

With limited substratum-facilitation and refuges from grazing (typically due to barnacles, which preferentially settle towards adult clumps) (e.g. Jenkins 2005), seaweeds may thus be less likely to establish on boulders. Though not statistically significant, the pattern observed in the Western site Ab may support this hypothesis (Fig. 2), which could be easily tested through limpet removals: likely due to extreme wave exposure in this site (Appendix S1, [START_REF] Duff | Les zones Natura 2000 de Corsen et Tréompan Vol 1: Les habitats de l'estran de la zone de Corsen[END_REF][START_REF] Robuchon | Multi-scale drivers of community diversity and composition across tidal heights: an example on temperate seaweed communities[END_REF], limpets were virtually absent from most boulders, presenting a featureless/smooth aspect and upon which ephemerals and young Fucus were observed in abundance. In NE Atlantic, it is often considered that the grazing pressure of limpets decreases lower on the shore [START_REF] Raffaelli | Intertidal Ecology[END_REF], as the ability of seaweeds to settle and grow (under decreasing stresses) becomes higher than limpet foraging. Indeed, the bedrock herein presented similar limpet densities between tidal heights, but the bare space was about sevenfold lower in the low-shore, where diverse assemblages of seaweeds and associated invertebrates were observed. In contrast, low-shore boulders were generally dominated by bare surface and only limpets and sheet-like ephemerals presented higher variability on boulders than on the bedrock at the scale of patch. Limpet abundance was up to 15-fold higher (× 3.9 on average per patch) on boulders than on the bedrock (and generally higher than on the midshore). Although this remains to be thoroughly tested, limpets when present may thus limit the establishment of a patchy seaweed assemblage, similar to surrounding bedrock. Interestingly, among the possible mechanisms regulating limpet abundance on boulders, habitat selection for substrate and food (e.g. biofilm growing on the apparent bare surface), survival due to lower predation and movement facilitated by bare surface [START_REF] Underwood | Effects of interactions between algae and grazing gastropods on the structure of a low-shore intertidal algal community[END_REF]) may all be promoted by disturbance. While physical disturbance (e.g. boulders knocking together, overturning) likely contributes to the contrasting patchiness observed between habitats, it may be either directly (by creating bare space) or indirectly (by modulating important processes, such as grazing).

Patterns and variability at increasing spatial scales

Beyond patches, variability patterns were generally consistent between boulder and bedrock with increasing spatial scales, i.e. from tens of meters to tens of kilometres. As an exception, the abundance of ephemerals varied equally among all spatial scales examined on the bedrock but showed a trend in decreasing variability on boulders as the horizontal span increases. This pattern tends to support the initial hypothesis that disturbance operating at smallscale will lessen other processes -visible on the bedrock -at larger scales. Comparable trends were however observed irrespective of the substrate for several groups (thick leathery, crustose) or solely on the bedrock (jointed calcareous) in the low shore.

The within-site variability (10s m) was generally similar to within-locality (1000s m), but a few contrasting patterns were observed in the low shore (e.g. jointed calcareous on the bedrock, barnacles and crustose on both substrata). Although contrasting between habitats and shore heights, some patch-to-patch differences were observed for the community structure, the richness and the abundance of all groups, expect sheet-like and crustose seaweeds (which displayed some site-to-site differences). Given the coastal features of the study sites (distributed along headlands or within inlets), hydrodynamics are likely to interact with substratum heterogeneity at these intermediate scales, hence influencing many aspects of recruitment and post-recruitment processes [START_REF] Blanchard | Scales of coastal heterogeneity: influence on intertidal community structure[END_REF].

Between the scales of locality to region (1-10s km), an increase in variability was observed for barnacle cover and community structure. These variables displayed differences, generally more pronounced on the bedrock than on boulders, between localities. On the bedrock, mid-shore communities ranged from fucoid-to suspension feeders-dominated, which could be due to gradients in wave exposure [START_REF] Hartnoll | Patchiness and fluctuations on moderately exposed rocky shores[END_REF][START_REF] Raffaelli | Intertidal Ecology[END_REF]. In the low shore, the splash and spray resulting from waves may allow the development of uplifted communities [START_REF] Raffaelli | Intertidal Ecology[END_REF], as herein observed westernmost location A (Fig. 3) probably undergoing greater swell than the others (Appendix S1). Should wave exposure be important in influencing bedrock communities among localities, it may also influence boulder communities, hence contributing to the similar variability patterns observed between habitats at this scale (α = 0.01). In addition, within a single patch and at a given tidal height, both boulders and bedrock could be expected to experience a similar propagule supply, either originating from local or remote sources. It could be hypothesized that the number of species likely to settle, colonize and eventually establish would tend to be similar between the bedrock and virtually non-disturbed boulders at local scale (Sousa 1979b[START_REF] Sousa | The responses of a community to disturbance: the importance of successional age and species' life histories[END_REF][START_REF] Chapman | Inconsistency and variation in the development of rocky intertidal algal assemblages[END_REF], although post-settlement survival may be influenced by a series of habitatand site-specific processes. Any discrepancy between biodiversity patterns of natural boulders and bedrock may thus emerge if the influence of these specific processes (e.g.. physical and biotic disturbance) prevails over those operating at larger scale. Across all studied patches, a positive correlation between boulders and bedrock richness was herein found on the low-shore, but not on the mid-shore (Fig. 5), and this may align with McGuinness (1987a) who found that the frequency and intensity of boulder overturning was generally higher on the shore. Although further work would be needed to disentangle putative mechanisms, disturbance is thus likely to prevail over a series of processes in influencing mid-shore boulders. Nonetheless, since boulder limpets may locally be excluded in favour of ephemerals' development (e.g. site Ab), the interaction between physical disturbance and biotic interactions may contribute to variability at large scales on boulders. In the low-shore, biodiversity patterns were generally more pronounced and involved a greater number of species than in the mid-shore. Through 'sampling effect', these patterns are more likely to co-vary between habitats, regardless of the disturbance (incl. grazing) seemingly stronger on boulders than on the bedrock. Like on the surrounding bedrock, all hydrodynamics, dispersal and recruitment processes may therefore contribute to large scale variability on boulders.

Beyond observations: implications and limitations

Overall, the present results suggest that several abiotic and biotic processes interact and may influence patterns at multiple spatial scales [START_REF] Levin | The problem of pattern and scale in ecology[END_REF][START_REF] Benedetti-Cecchi | Testing the relative contribution of positive and negative interactions in rocky intertidal communities[END_REF], Dal Bello et al. 2017), even within two distinct microhabitats. While habitat properties may locally affect diversity and assemblages, such relationships are generally mediated, over a range of spatial and temporal scales, by the biological and environmental contexts [START_REF] Tews | Animal species diversity driven by habitat heterogeneity/diversity: the importance of keystone structures[END_REF][START_REF] Matias | Macrofaunal responses to structural complexity are mediated by environmental variability and surrounding habitats[END_REF][START_REF] Leclerc | Habitat formation prevails over predation in influencing fouling communities[END_REF]. In the studied system, wave action and limpet grazing may have conspicuous influence on boulder field communities. Nonetheless, experimental manipulations would be needed to infer these processes and to define the spatial scales at which they operate (e.g. [START_REF] Benedetti-Cecchi | The interplay of physical and biological factors in maintaining mid-shore and low-shore assemblages on rocky coasts in the north-west Mediterranean[END_REF]. While the present study helps direct future research in boulder fields, its outcomes may depend on the specific temporal frame at which it was conducted [START_REF] Benedetti-Cecchi | Variability in abundance of algae and invertebrates at different spatial scales on rocky sea shores[END_REF][START_REF] Maggi | Spatio-temporal variability in Mediterranean rocky shore microphytobenthos[END_REF]. First, the intensity and the spatial extent of physical and biotic processes may vary over time [START_REF] Osman | The establishment and development of a marine epifaunal community[END_REF]. The present study was performed in late fall, when tourism and associated recreational fishing are reduced, as compared to spring or summer holiday periods. Furthermore, the intensity of recreational fishing is unlikely to be even from site to site [START_REF] Addessi | Human disturbance and long-term changes on a rocky intertidal community[END_REF]). As such, the balance between small-and intermediate-scale variability may be time-dependent. Should they contribute to physical disturbance, strong storms -operating at the scale of the region, though probably interacting with habitat heterogeneity at small scale -are less likely to occur during summer. Second, the nature and abundance of species interacting with these processes also likely change over time. For instance, most fleshy red algae grow and reproduce during the winter-spring period (Sousa 1979b), i.e. a few months after the sampling was performed. Unlike ephemeral algae (reproducing all the yearlong), these middle and late successional species could have displayed very different patterns of variability later in the year. Finally, since all natural and anthropogenic disturbances may vary throughout the year, incorporating a temporal level of variability would have been even more valuable than simply replicating the survey [START_REF] Smale | Multi-scale patterns of spatial variability in sessile assemblage structure do not alter predictably with development time[END_REF], Leclerc et al. 2016). Given the spatially patchy nature of both boulders and bedrock, such direction may be promising to define indices of anthropogenic disturbance on the basis of variability [START_REF] Warwick | Increased variability as a symptom of stress in marine communities[END_REF][START_REF] Fraschetti | Spatial variability and human disturbance in shallow subtidal hard substrate assemblages: a regional approach[END_REF][START_REF] Chapman | Patterns of spatial and temporal variation of macrofauna under boulders in a sheltered boulder field[END_REF]) -an approach which has so far yielded conflicting results (e.g. [START_REF] Chapman | Variability at different spatial scales between a subtidal assemblage exposed to the discharge of sewage and two control assemblages[END_REF]. Nonetheless, variance analyses generally rely on independency and such temporal replication would be challenged by the limited number of boulders available per sampling units and risks of repetition should be considered.

CONCLUSIONS

In conclusion, important patchiness was revealed on either boulders or 'stable' bedrock, though generally higher on the bedrock than on boulders, except regarding the abundance of ephemeral algae. While the degree of spatial variability in this group could be useful to characterize physical disturbance on boulders, it is unlikely to disentangle between natural and anthropogenic causes from one shore to another, and even more likely to interact with additional sources of disturbance such as grazing, especially by limpets. Like within most rocky shores worldwide, intertidal communities of NW Brittany -regardless of substrate type -experience important sources of variability across multiple spatial scales (100s cm, 10s m, 1000s m and 10s km) which should be considered in sampling designs for both experimental and monitoring purposes.

  ') were limited to the bedrock in the low-shore (A > B = C = D), in the midshore (A = (B = C < D)) and to boulders in the mid-shore (A = B = C < D).

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. Univariate patterns in richness and abundances of various groups in mid-shore (left panels) and low-shore (right panels). Mean values (± SE, n = 16) for richness (A, A'), numerical abundance of Patella spp. (B, B'), covers of barnacles (C, C'), bare space (D, D'), ephemeral (E, E'), filamentous (F, F'), coarsely branched (G, G'), canopy (H, H'), jointed calcareous (I, I') and crustose algae (J, J') are indicated for each habitat with patch.

Figure 3 .

 3 Figure 3. Principal coordinate analyses describing mid-shore (A, B) and low-shore assemblage structure (C, D). Vector plots of variable correlated with the PCO axes are indicated on right panels (r > 0.4), with font varying with r values (see legend in the figure).

Figure 4 .

 4 Figure 4. Pseudo-variance estimates (mean ± SE, n = 8, logarithmic scale) in various multivariate and univariate descriptors of the assemblages depending on habitats and shore levels within four spatial scales (Patch, Site, Locality and Region).

Figure 5 .

 5 Figure5. Relationship between the observed combined richness on boulder and on the bedrock per patch in the mid-and in the low-shore (i.e. total number of species identified within a total of 16 quadrats). Only significant regression (within low shore) is depicted.

  Fig.1

Acknowledgements

. I thank D Davoult, G Schaal and S Huchette for suggesting potential boulder-fields in the region and L. Lévêque for sharing corresponding maps. I am grateful to M Dal Bello, A Brante and M Robuchon for their insightful feedback on an earlier draft as well as three anonymous reviewers for their valuable comments that helped to improve the manuscript. This work has been funded by 'Pôle Emploi' and a postdoctoral fellowship CONICYT FONDECYT/POSTDOCTORADO/3160172.

Table 1. Results of ANOVA tests for differences in multivariate and univariate pseudo-variance estimates across habitats, levels and spatial scales. **: P < 0.01, ***: P < 0.001, a : term not significant at α = 0.01, level adopted given the heteroscedasticity of the data.


Amisha Maharaja

Andre Journel

Guillaume Caumon

Sebastien Strebelle

Assessment of Net-To-Gross Uncertainty at Reservoir Appraisal Stage: Application to a Turbidite Reservoir Offshore West Africa

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The quest for new hydrocarbon reservoirs and technical advances take the industry deeper offshore, opening new domains for exploration and appraisal. In such a context, due to the high cost of reservoir development, a reliable uncertainty assessment of the original hydrocarbon in place (OHIP) is a must to decide on the economic viability of a field.

Due to high cost of drilling in offshore environments, wells tend to be sparse and preferentially located in high pay zones. Assessing the uncertainty from such sparse data requires identifying and ranking the major parameters that control the OHIP uncertainty, and defining a sound method to make these parameters variable. Geostatistical simulation methods [START_REF] Goovaerts | Geostatistics for natural resources evaluation[END_REF][START_REF] Chilès | Geostatistics: Modeling Spatial Uncertainty[END_REF][START_REF] Lantuejoul | Geostatistical Simulation. Models and Algorithms[END_REF] do not produce a good measurement of global uncertainty. Indeed, the variations resulting from such techniques are fluctuations about some global target distribution, which typically are insignificant compared to the uncertainty due to poorly constrained reservoir volume and geological scenario.

Global Net-to-Gross (NTG) of a reservoir is a major source of OHIP uncertainty during exploration and appraisal. In case of turbidite reservoirs, NTG ratio is essentially the proportion of sand in the reservoir. Various techniques have been proposed to assess this uncertainty from the classical Bootstrap technique [START_REF] Efron | Bootstrap methods: Another look at the jackknife[END_REF], to the Dirichlet distribution approach [START_REF] Biver | An overview of different techniques to quantify uncertainties on global statistics for geostatistical modeling[END_REF][START_REF] Biver | Uncertainties in facies proportion estimation, ii. application to geostatistical simulation of facies and assessment of volumetric uncertainties[END_REF][START_REF] Haas | Uncertainties in facies proportion estimation, i. theoretical framework the dirichlet distibution[END_REF], to Spatial Bootstrap [START_REF] Journel | Resampling from stochastic simulations[END_REF][START_REF] Norris | Early quantification of uncertainty in the estimation of oil-in-place in a turbidite reservoir[END_REF][START_REF] Bitanov | Uncertainty in n/g ratio in early reservoir development[END_REF].

More recently, a novel methodology for global NTG uncertainty assessment was proposed by [START_REF] Caumon | Reflections on early uncertainty assessment: Application to a hydrocarbon reservoir development[END_REF], and further developed by [START_REF] Maharaja | Global Net-to-Gross uncertainty assessment at reservoir appraisal stage[END_REF]. This methodology overcomes some of the limitations of the previous approaches. It builds on spatial bootstrap by recognizing that the two key input parameters for generating stochastic realizations, namely the geological scenario and the global NTG, are both uncertain, hence randomized. Geological scenario, G, is randomized by retaining all alternate depositional models that fit the available data (Figure 1, Panel A). Next, the global NTG value A associated with each geological scenario is randomized by considering a distribution of global NTG values, P(A|G), which specifies both the range and prior probability of NTG values that a particular scenario can take. The prior NTG distribution can be obtained by pooling together NTG information from analog fields and outcrops studies. The goal of the workflow is to update this prior NTG distribution given actual observed data, D = d 0 . To perform this update, the prior NTG distribution is first discretized into M classes. For each class, a representative facies realization is simulated from the corresponding geological scenario, while honoring all available data (Figure 1, Panel B). Synthetic data sets are re-sampled from this realization by following the initial drilling strategy. The NTG estimates computed from these synthetic data sets provide the likelihood of observing the initial NTG estimate in a simulated reservoir with that global NTG (Figure 1, Panel C). Then using Bayes rule, the prior probability of the global reservoir NTG being in any given class m is updated into a posterior probability given the observed NTG estimate. Repeating this procedure for all prior global NTG classes and pooling together the posterior probability of each NTG class provides the posterior probability distribution of global NTG (Figure 1, Panel D). This distribution is a model of uncertainty specific to the reservoir under study and for a particular geological scenario. Posterior distributions corresponding to different geological scenarios can be combined into a single posterior, but this calls for a prior probability of occurence of each scenario.

The reader is referred to [START_REF] Maharaja | Global Net-to-Gross uncertainty assessment at reservoir appraisal stage[END_REF] for a detailed description of the methodology and a comprehensive presentation of its application to a turbidite reservoir offshore West Coast of Africa, hereby called the WCA reservoir. In this paper, the key results of the WCA reservoir case study are presented to demonstrate the practicality of this method. The information available at appraisal stage is presented in the next section.

WCA RESERVOIR DATA SET

Geological scenarios

The seismic data quality for the WCA reservoir is sufficiently good to identify it as a part of multiple slope channel complexes, or slope valley system. The depositional facies filling the slope valley, however, cannot be easily inferred from the seismic data. This facies uncertainty is expressed through the three different training images (Ti) shown in Figure 2. A Ti is a necessary input to multiple-point simulation (mps) algorithms (Remy et al., in press). It is a repository of types of structures deemed present in the reservoir. It must depict not only the shapes and dimensions of all facies, but also how these facies interact with one another [START_REF] Maharaja | Tigenerator: Object-based training image generator[END_REF]. In this case study, the snesim algorithm is retained to generate the conditional geological scenarios [START_REF] Strebelle | Conditional simulation of complex geological structures using multiple-point statistics[END_REF]. Four depositional facies with varying NTG ratios are identified from core analysis. In Ti1 and Ti2, which represent actual scenarios for the WCA reservoir, facies 1

is the poor quality background shaly facies and facies 4, which is interpreted as sand-filled channels, is the good quality reservoir rock. The two scenarios differ in interpretation of facies 2 and 3, which are both intermediate quality facies. In scenario 1, both facies 2 and 3 are classified as channels, while in scenario 2 they are considered to be lobate deposits. In Ti3, which is a hypothetical scenario retained to study the impact of geological scenario, facies 2, 3 and 4 are in form of lobes.

Prior probability distribution for global net-to-gross

A prior probability distribution of global NTG for the WCA reservoir is obtained from a database of analog reservoirs (Figure 3). Only one prior NTG distribution is available for the three different scenarios, because the database used to obtain this distribution classifies reservoirs at system scale (here slope valley), not at the reservoir facies scale.

Well and seismic data

The four wells available at appraisal stage are shown in Figure 4(a). In reality, the WCA reservoir has been in production for a few years and 28 wells are available, providing a good coverage of the reservoir (Figure 4(b)). The wells are marked with the order in which they were drilled, which makes it possible to study the impact of additional data on NTG uncertainty.

As mentioned earlier, four different facies with varying NTG ratios (sand proportion) are determined from cores, see Table 1. Equation 1 is used to compute the NTG by combining the facies proportions from the wells p i and the within-facies sand proportion n i given in Table 1 under the constraint that the proportions sum to one. The NTG estimate from the four appraisal wells is 0.484. Facies number 1 2 3 4 NTG value, n i (%) 3.8 47.3 63.0 90.7

a ⋆ = ∑ i (p i × n i ) ; with ∑ i p i = 1 (1)
The four appraisal stage wells are clustered in the thicker eastern part of the WCA reservoir, hence the NTG estimate obtained from these wells may not be representative of the whole reservoir. Indeed, integrating acoustic impedance data with the four wells lowers the global NTG estimate to 0.436. Bayes inversion, as described in [START_REF] Bitanov | Uncertainty in n/g ratio in early reservoir development[END_REF], is used to calibrate the impedance data. Many different techniques have been proposed to calibrate seismic data with well data [START_REF] Fournier | A statistical methodology for deriving reservoir properties from seismic data[END_REF][START_REF] Goovaerts | Geostatistics for natural resources evaluation[END_REF][START_REF] Caers | Modeling conditional distributions of facies from seismic using neural nets[END_REF][START_REF] Strebelle | Modeling of a deepwater turbidite reservoir conditional to seismic data using multiple-point geostatistics[END_REF]; the workflow, however, does not call for any particular technique. Seismic data calibration is a major source of uncertainty at appraisal stage, yet checking robustness of the calibration with cross-validation is difficult due to data sparsity. The uncertainty in NTG estimate due to calibration errors can be assessed by randomizing the parameters of a particular algorithm or by using different algorithms to obtain alternate NTG estimates. Refer to [START_REF] Maharaja | Global Net-to-Gross uncertainty assessment at reservoir appraisal stage[END_REF] for a discussion on how to incorporate different initial NTG estimates into the workflow.

During step 2 of the workflow (Figure 1, Panel B), the global NTG value (i.e. the average sand proportion) is known, but we need to know the proportion of each facies corresponding to that global NTG value in order to simulate a facies realization for that class. In a two facies case, Equation 1 is sufficient to compute the facies proportions because we have two unknowns (the two facies proportions) and two constraints (Equation 1 and ∑ i p i = 1). In the WCA reservoir there are four facies; hence, two additional constraints are necessary to solve for the four unknown facies proportions.

One way to introduce additional constraints is to freeze the ratio of facies proportions. These facies proportions ratio are uncertain parameters; they can be obtained from analog knowledge. This approach works best when facies proportions are tightly related. In case of the WCA, we froze the ratio of proportions of facies 2 to facies 4 and that of facies 3 to facies 4. These ratios for each scenario were taken from the corresponding Ti. The proportion of the four facies in the three scenarios are similar.

DISCUSSION

Any NTG uncertainty model obtained using this workflow is tied to the sampling domain. The reservoir volume, however, was frozen for this study. A joint randomization of volume and NTG is a top consideration for future work. Reservoir volume uncertainty is linked to uncertainty about seismic data processing and interpretation. Time-to-depth conversion is one of the largest sources of uncertainty about the seismic data because of the poorly understood velocity models required for this conversion. The uncertainty about reservoir volume and NTG can be jointly assessed by carrying out the NTG workflow in alternate reservoir volumes resulting from widely different velocity models or different horizon interpretations.

This workflow was applied to a deepwater turbidite reservoir with sparse appraisal stage wells. However, it is general enough to be applied on any depositional environment with any number of facies. Training image-based mps simulation was used to generate the geological scenario, however, any simulation technique that honors all available data, including seismic, and adequately reproduces geological scenario and the target NTG can be used. In case of sparse, preferentially located wells it is especially important to correct any bias in NTG estimate by incorporating seismic data. Calibrating seismic data to wells is a major source of uncertainty and should be accounted for by randomizing the parameters of a particular calibration algorithm or by using different algorithms to obtain alternate NTG estimates.

Results with the four appraisal wells showed that the posterior distribution became narrower as more wells were introduced. This is because the geological scenario and the prior NTG distribution were the same. If the additional well drastically changes the geological scenario, then the prior distribution should be revised. If the revised prior distribution is wider than the previous one, then the posterior distribution may be wider as well, implying more uncertainty.

From a value of information perspective, before drilling a new well we can study its impact on reduction of NTG uncertainty by carrying out the workflow with different well locations. The posterior distributions depend on the size and distribution of the heterogeneities and the configuration of the wells that sample them.

The workflow can also be applied to assess uncertainty about continuous variables, such as average porosity of a reservoir. In this case, the prior distribution provides the range that average porosity can take based on historical information. For a given class of global mean porosity, additional assumptions must be made regarding the shape of the distribution to perform porosity simulations. This approach provides more informed models of global porosity uncertainty than simulating multiple realizations using the same global porosity distribution.

As with any uncertainty assessment, each posterior distribution is one model of global NTG uncertainty for the WCA reservoir, that depends completely on the various assumptions and decisions that went into building the model. It would be incomplete without a clear statement enlisting all these assumptions and decisions.

Figure 1 :

 1 Figure 1: Workflow for global net-to-gross uncertainty assessment.

Figure 2 :

 2 Figure 2: Three facies scenarios retained to study the impact of geological scenario.

Figure 3 :

 3 Figure 3: Prior global net-to-gross probability distribution for the WCA reservoir obtained from a database of analog reservoirs.

Figure 4 :

 4 Figure 4: Location of all available wells, and of the four appraisal wells (vertical exag. ×3).

  Ideally, each of the alternative geological scenarios retained should have a different prior NTG distribution. The goal of the workflow is to update this prior distribution given the actual quantitative data from the WCA reservoir.

		Prior NTG distribution
					mean: 0.337
					std. dev.: 0.124
					minimum: 0.060
					P10: 0.180
					median: 0.320
					P90: 0.520
				maximum: 0.610
	Frequency
	0.060	0.198	0.335	0.473	0.610
			Value

Table 1 :

 1 Within-facies sand proportion for the four facies in the WCA reservoir

ACKNOWLEDGEMENTS

The authors would like to acknowledge the Chevron Energy Technology Company for supporting this research.

RESULTS OF GLOBAL UNCERTAINTY STUDY

Impact of geological scenario

The four appraisal stage wells and a NTG estimate of 0.436 are retained for all three scenarios. All parameters except for the geological scenario are similar for the three cases. Refer to [START_REF] Maharaja | Global Net-to-Gross uncertainty assessment at reservoir appraisal stage[END_REF] for detailed information about the workflow parameters used in the WCA study and their sensitivity analysis. The posterior distributions conditional to this estimate are given in Figure 5. The mean and median of the three posteriors are comparable. However, the width of the [p 10 , p 90 ] interval changes with the scenario. This confirms the fact that geological scenario matters. Hence, if a geological scenario is uncertain, which is likely at appraisal stage, all alternate scenarios that fit the data should be considered.

The [p 10 , p 90 ] interval corresponding to scenario 1 (all channels) is the narrowest. The p 10 of both scenario 2 (channels and lobes) and scenario 3 (all lobes) is lower than that of scenario 1, because lobes, as opposed to channels, increase the likelihood that the four wells are either all contained inside or outside the lobes. Morever, in scenario 2 the high NTG channel facies is thicker than in scenario 1 and the channels tend to be more clustered. Scenario 2 is a hybrid between scenarios 1 and 3, so it is consistent that its posterior statistics fall between those of the two more extreme scenarios. It is important to freeze the well configuration when resampling from stochastic realizations, especially in the case of sparse, clustered wells.

Impact of prior distribution

As mentioned earlier, only one prior probability distribution for global NTG is available for the three facies scenarios. To study the impact of prior probability distribution, we retain two additional distributions -a triangular distribution between [0.25, 0.65] with a mode 0.50 and a uniform distribution over [0.25, 0.65]. These distributions are narrower than the distribution obtained from the analog database (Figure 3). Only scenario 3 (all lobes) is retained for this study. Figure 6 compares the posteriors resulting from using the triangular and uniform prior distributions. Note that the initial NTG estimate, 0.44, is the same in both cases because the same set of four initial wells and acoustic impedance data are retained. Hence, the differences in the posterior distributions is solely due to the different prior probabilities. Both posteriors are narrower than the one resulting from using the analog prior distribution (Figure 5(c)), since the triangular and uniform prior distributions are narrower than the analog prior. Note that the posterior distribution cannot be wider than prior distribution because the prior probability of any NTG value that is outside the bounds of the prior distribution is zero, hence the posterior probability of that NTG value will also be zero according to Bayes rule.

Impact of additional data

The impact of additional data on the posterior NTG distribution is studied using the channel scenario 1, which is considered the mostly likely scenario by the geologist. The goal is to see how the posterior NTG distribution changes as additional well data become available. We repeat the entire workflow four times, first using only well 1, then adding well 2, then well 3 and finally with all four appraisal stage wells. The only parameters that change from one run to another are the number of wells and the corresponding initial NTG estimates. The same acoustic impedance cube is used in all four runs, but the calibration is repeated as new data becomes available. Table 2 summarizes the well NTG and the updated NTG estimates after integration of acoustic impedance data. The mean and [p 10 , p 90 ] of the corresponding posterior distributions are also given. The [p 10 , p 90 ] interval of the posterior distribution with the first well is much narrower than that of the prior NTG distribution [0.18, 0.52] (Figure 3). The posterior distribution mean and the [p 10 , p 90 ] interval shifts to the right when well 2 is introduced because the combined NTG estimate of wells 1 and 2 is much higher than that of well 1. Also, the [p 10 , p 90 ] interval of the posterior with two wells is narrower than that with well 1 only. This trend is also observed when wells 3 and 4 are introduced. In all four cases the [p 10 , p 90 ] interval is roughly centered over the corresponding posterior mean as expected. The initial NTG estimates ( For verification purposes, the workflow is run with all available 28 wells using the channels scenario and the analog prior distribution. NTG estimate from the 28 wells is 0.43; using seismic data lowers this estimate to 0.41, a small reduction since the 28 wells sample the reservoir adequately (Figure 4(b)). The means of the two posteriors are similar, however, the [p 10 , p 90 ] interval of the posterior with 28 wells [0.36, 0.42] is significantly narrower than the [p 10 , p 90 ] resulting from using the four appraisal wells [0.32, 0.44]. This is expected, because with 28 wells the global NTG is closely estimated.

Nb wells