Marie-Anne Félix
email: felix@ijm.jussieu.fr

Keywords: population genetics, Caenorhabditis elegans, temporal survey, outbreeding depression

Caenorhabditis elegans is a major laboratory model system yet a newcomer to the field of population genetics, and relatively little is known of its biology in the wild. Recent studies of natural populations at a single timepoint revealed strong spatial population structure and suggested that these populations may be very dynamic. We have therefore studied several natural C. elegans populations over time and genotyped them at polymorphic microsatellite loci. While some populations appear to be genetically stable over the course of observation, others seem to go extinct, with full replacement of multilocus genotypes upon regrowth. The frequency of heterozygotes indicates that outcrossing occurs at a mean frequency of 1.7% and is variable between populations. However, in genetically stable populations, linkage disequilibrium between different chromosomes can be maintained over several years, at a level much higher than expected from the heterozygote frequency. C. elegans seems to follow metapopulation dynamics, and the maintenance of linkage disequilibrium despite a low yet significant level of outcrossing suggests that selection may act against the progeny of outcrossings.

INTRODUCTION

Most population genetic studies infer evolutionary mechanisms of a population from a single timepoint. There are as yet few molecular studies of the same population over time [START_REF] Viard | Population dynamics inferred from temporal variation at microsatellite loci in the selfing snail Bulinus truncatus[END_REF][START_REF] Guillemaud | Spatial and temporal genetic variability in French populations of the peach-potato aphid, Myzus persicae[END_REF][START_REF] Meunier | Small effective population sizes in a widespread selfing species, Lymnaea truncatula (Gastropoda: Pulmonata)[END_REF][START_REF] Charbonnel | A long-term genetic survey of an ungulate population reveals balancing selection acting on MHC through spatial and temporal fluctuations in selection[END_REF][START_REF] Trouvé | Ecological components and evolution of selfing in the freshwater snail Galba truncatula[END_REF]. However, to get a direct picture of dynamic phenomena such as migration, recombination, selection or population extinction, and to detect variation over time, temporal surveys of populations are required. Here we present a temporal study of several natural populations of the nematode Caenorhabditis elegans over three years.

Beyond the fact that C. elegans is a major laboratory model organism with a fast generation time (3.5 days in standard laboratory conditions), an interesting feature for evolutionary biology is its peculiar mode of reproduction: C. elegans has two sexes, selfing XX hermaphrodites and facultative XO males that are able to mate with hermaphrodites.

Males arise either spontaneously by rare non-disjunction of X chromosomes at meiosis (at a rate around 0.1%; [START_REF] Hodgkin | Natural variation and copulatory plug formation in Caenorhabditis elegans[END_REF][START_REF] Teotónio | Genetic variation for outcrossing among Caenorhabditis elegans isolates[END_REF] or as progeny of hermaphrodites when mated with males (50% of the cross progeny is male). This facultative outcrossing makes C. elegans an excellent system to study the impact of outcrossing in a diploid organism.

Studies on natural populations of C. elegans have only recently begun. This species displays low overall levels of polymorphism (similar to humans, but 20-fold lower than Drosophila melanogaster) and displays only weak geographic structure at a worldwide scale [START_REF] Koch | Single in wild isolates of Caenorhabditis elegans[END_REF][START_REF] Denver | Phylogenetics in Caenorhabditis elegans: an analysis of divergence and outcrossing[END_REF][START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF][START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF], but see HABER et al. 2005). Selfing is clearly the predominant mode of reproduction in the wild, but outcrossing rate estimates range from 0.01%, to 1-20%. The lower number (0.01%) was estimated from studies of linkage disequilibrium, either within local populations [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF] or among a worldwide set of isolates [START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF]. The higher range was estimated from measures of heterozygote frequencies in populations from France (1.3%;[START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF] and Los Angeles (20%;[START_REF] Sivasundar | Sampling from natural populations with RNAi and population structure in Caenorhabditis elegans[END_REF].

The habitat of C. elegans on ephemeral resources and its population genetic structure led to the suggestion that it may follow metapopulation dynamics [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF][START_REF] Sivasundar | Sampling from natural populations with RNAi and population structure in Caenorhabditis elegans[END_REF], with populations frequently going extinct and habitats being recolonized through migration [START_REF] Hanski | Metapopulation ecology[END_REF]. However, this was suggested by indirect evidence, and a temporal study of local populations has so far been lacking. We have thus followed C. elegans populations in several locations over the span of one to three years, with the goal of identifying the population dynamics shaping its evolution. We found that some C. elegans populations were ephemeral, and observed metapopulation dynamics, with extinction followed by recolonization by new genotypes. Surprisingly, in the largest and most stable population, genetic linkage between the same alleles persisted over three years despite a detectable rate of outcrossing, suggesting selection acting against the progeny of a recombination event. The observed metapopulation structure and the maintenance of linkage disequilibrium may explain the discrepancy between outcrossing rates measured at short and long timescales through heterozygote frequency and linkage disequilibrium, respectively. We finally discuss how the observed dynamics of natural C. elegans populations may influence the genetic and phenotypic evolution of this species.

MATERIALS AND METHODS

Sampling

The sampling procedure and most locations were described in [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF]. New locations include Obernai, Bas-Rhin, France (position: 48.46°N, 7.48°E), leaf litter next to a vegetable garden in the middle of vineyards, and the Botanical garden of the University of Lisbon,Portugal (38.42°N,9.12°W), leaf and fruit litter below trees and in a compost heap. Samples were collected from September 2002 to January 2006 (details on sampling can be found in Table S1).

Sampled material was spread on standard NGM plates seeded with E. coli OP50.

Worms were picked within 1 hour to 2 days after plating; the developmental stage was recorded, as described in [START_REF] Barrière | Isolation of C. elegans and related nematodes in WormBook[END_REF]. On several occasions, six samples were taken a few centimeters apart in a single compost pile.

In most cases, individuals sampled from the soil were left to develop on the plates and self-progeny of hermaphrodites were harvested. One portion was frozen in glycerol and kept at -80°C while the rest was lysed in Worm Lysis Buffer (a digestion buffer commonly used to prepare DNA from C. elegans: 50 mM KCl, 10 mM Tris pH 8.2, 2.5 mM MgCl 2 , 0.45% NP-40, 0.45% Tween-20, 0.01% gelatin, 0.06% proteinase K), the lysate conserved at -20°C and used for subsequent amplification. This procedure conserves the genotype of the collected individual, even when heterozygous (labeled 'heterozygous' in Table 2). In other cases (labeled 'inbred' in Table 2), isogenic strains were established by selfing for several generations [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF], and possible heterozygosity was lost.

Microsatellite genotyping

Six microsatellite regions defined in [START_REF] Haber | Evolutionary History of Caenorhabditis elegans Inferred from Microsatellites: Evidence for Spatial and Temporal Genetic Differentiation and the Occurrence of Outbreeding[END_REF] were amplified by PCR. The forward primer was labeled with a fluorophore, either Hex or 6-Fam, and amplified fragments were run on a ABI 3100-avant system. Primers were: II-L: f: AACAAAAATGTGGCAGGGAG, r: GGGTTACGGTAGTGGTACTGTAGG. III-R: f: GATGAATGGATATGACCGGC, r: TATCAGGCGTATCACCTCCC.

IV-L: f: AAGATTTCTGCTAACGTGCTGA, r: AGTAACTTTGGTGCAGGTTCG.

V-L: f: CGTTGGGACAGGATCTAGTTG, r: CGTTGGGACAGGATCTAGTTG.

X-R: f: GCACACGCTTGAATGTCATAA, r: AAGAGCAGTAGCCGTTGTTGA.

For the II-R locus, we used a slightly different protocol, with the forward primer tailed with an M13 sequence, and amplification being conducted with a labeled M13 primer.

Primers were: f: CACGACGTTGTAAAACGACTTCTCATTGGAAAGTTGGGC, r: CAATACCGAGAAACGGATGAA.

When a putative heterozygote was found, the glycerol stock was thawed, revived and individual worms were genotyped to check whether both alleles segregated. Microsatellite repeat numbers were deduced from PCR fragment length and comparison with the repeat number in N2.

For microsatellite locus II-L in samples from Le Perreux-sur-Marne, we had reproducibility problems, and therefore excluded this locus from further analysis in this population. In several isolates from the Lis12-0705 sample, genotyping at locus IV-L revealed two different fragment sizes, corresponding to 35 and 44 repeats. This polymorphism did not segregate in the self progeny as would be expected from a heterozygote at a single locus, and the progeny of a cross with N2 males displayed three allele sizes (that of N2 plus the two others from Lis12-0705), suggesting a duplication of this locus in Lis12-0705. Since one individual (LisbonP12D3, Dataset S1) displayed only the 35-repeat allele, we considered this 35-repeat allele to be at the locus IV-L genotyped in other strains. Similarly, one individual from the Lis14-0705 sample displayed two fragment sizes, corresponding to 36 and 46 repeats; we considered the 36-repeat allele to be at the locus orthologous to IV-L.

Data analysis

The pairwise difference, the scaled mutation parameter Θ hom and gene diversity H were calculated with Arlequin V. 3.01 [START_REF] Excoffier | Arlequin (version 3.0): An integrated software package for population genetics data analysis[END_REF], over all loci. Θ hom derives from gene diversity by the relationship H= Θ hom Θ hom +1 . Population structure, as measured by θ [START_REF] Weir | Estimating F-statistics for the analysis of population structure[END_REF], an estimator of F st , and 95% confidence intervals were calculated by bootstrap with FSTAT (GOUDET 2001).

(
The best statistics to test for population differentiation with unbalanced samplings is not F st or its components, but the likelihood ratio G statistic [START_REF] Goudet | Testing differentiation in diploid populations[END_REF].

Differentiation between samples was tested in R (R DEVELOPMENT [START_REF] Development | R: a language and environment for statistical computing[END_REF] with the package hierfstat (GOUDET 2005).

To calculate the inbreeding coefficient f, [START_REF] Weir | Estimating F-statistics for the analysis of population structure[END_REF], an estimator of F is , we used GDA [START_REF] Lewis | Genetic Data Analysis: Computer program for the analysis of allelic data[END_REF]. Its confidence interval was obtained by bootstrapping over loci. Selfing rate was calculated as s=2f/(1+f).

Genotypes were obtained either on the pooled progeny of an individual sampled from the wild ('non-inbred'), or after inbreeding for a few generations in the laboratory by picking a single hermaphrodite individual ('inbred' strains, designated with a JU strain number). This feature is indicated for each sample in the second column of Table 2. For calculations involving comparisons between laboratory inbred and non-inbred genotypes, we considered inbred strains as non-inbred that are homozygotes at all loci: given the very high level of inbreeding witnessed in non-laboratory-inbred populations, this can be considered a reasonably good approximation.

In the case of non-inbred genotypes, we deduced haplotypes from genotypes: in most cases, they were homozygous; for the five individuals that were heterozygous at two or more loci, we inferred the phase from other haplotypes found in the same sample.

The standardized index of association I A S (multilocus measure of linkage disequilibrium) and its significance (p-value) were calculated for the different samples with Lian v. 3.5 [START_REF] Haubold | LIAN 3.0: detecting linkage disequilibrium in multilocus data[END_REF], using the parametric test.

For the confidence intervals on linkage disequilibrium in Franconville, values of D'

were calculated from data, and the expected genotype frequency was calculated, based on allele frequencies, for gradually more distant values of D'. The concordance of these expected tables with the observed data was then calculated by a polynomial probability estimate.

Pairwise comparisons between two timepoints were carried with the same method, calculating the concordance of D' values between the two samples. The R function used is available from the authors at request.

For estimations of generation times compatible with the observed linkage disequilibrium over time, the latter was considered to decay according to D' N+1 =(1-r)D' N , N being the generation number, r being the recombination rate. At generation N, linkage disequilibrium would be D' N =(1-r) N D' 0 . From linkage disequilibrium, it is thus possible to calculate N as N=ln(D' N /D' 0 )/ln(1-r). However, in C. elegans, r is diminished by inbreeding and lower than the normal recombination rate. Therefore, the observed recombination rate is

r'=r(1-F is ) (NORDBORG 1997). Thus, N=ln(D' N /D' 0 )/ln(1-r(1-F is )).

Embryonic lethality

To check for possible incompatibility between genotypes, embryonic lethality and brood size were monitored in the F2 progeny of interstrain crosses (JU360 males with JU361 hermaphrodites, and the reverse cross). F1 hermaphrodites were picked at the L4 stage, and transferred every 8 hours to a new plate, until sperm exhaustion. 24 hours after transfer, unhatched eggs were counted, and a further 24 hours later, larvae were counted and abnormal phenotypes recorded. After they had finished laying, F1 hermaphrodites were genotyped at the II-R locus, to differentiate self from cross-progeny. Self progeny provided an internal control.

RESULTS

We first outline the sampling structure of the natural C. elegans populations that we followed, describing their habitat, developmental stage and density fluctuations. We then turn to the microsatellite genotypes of isolated individuals, first analyzing the molecular diversity and heterozygote frequency in each locality, and the spatial structure at different scales. We finally analyze the temporal dynamics of these populations and the dynamics of linkage disequilibrium.

Habitat and population density

Samplings: We sampled C. elegans in different locations in France and Portugal, including those of our previous study [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF] (Figure 1A). In several instances, samples were collected at different points within a location to probe for population structure at a small spatial scale (Figure 1B-D). In addition to compost heaps, we sampled rotting fruits in the same gardens and in the Botanical Garden in Lisbon, Portugal.

The localities of our previous study had been sampled either in 2002 or in 2004 [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF]. Among those, the Franconville, Le Perreux, Le Blanc and Hermanville compost heaps were newly sampled at frequent intervals until January 2006.

Only the two former yielded C. elegans every time (Franconville) or at most timepoints (Le Perreux). For the two latter, we could only find C. elegans again once or twice in 2005. We also resampled in 2005 the other localities of our previous study (Merlet, Primel, Sainte-Barbe). Each sample was named using the first letters of the location followed by the month and year of sampling (e.g. Fra-1102 for the November 2002 sampling of Franconville)

(Tables 1 andS1). Given the very high selfing rate, a population is difficult to define for C.

elegans. For the sake of simplicity, we herein refer to all C. elegans individuals from a single location (e.g. a single compost pile) as a population.

Habitat and developmental stage: Our sampling procedure allowed us to isolate all C. elegans individuals from a sample, most of them within a few hours [START_REF] Barrière | Isolation of C. elegans and related nematodes in WormBook[END_REF]. In addition to previously described habitats (compost heaps, soil and snails), we found C. elegans in rotting fruits fallen below their trees: apples, figs, tomatoes, plums, pears, fruits from a Ficus tree and from an unidentified tree in the Lisbon Botanical Garden. Stages other than dauer were found mainly in fresh compost and decaying fruits, in samples Bla-1105, HerF-1005, Lis12-0705 (Table 1).

Density at a given time: Overall, density was higher in fresh compost and decomposing fruits than in older compost or soil. The highest density was found in fresh compost in Pri-0805 (21 individuals/gram; Table S1). Within the compost heaps where several samples were taken at the same time, strong variations in C. elegans concentration could sometimes be found at a small scale (Table S1, Figure S1).

Density over time:

Given the large variance observed between samples collected on the same day, it may be hazardous to draw a strong conclusion on temporal dynamics using timepoints when a single sample was collected. However, in Franconville and Le Perreux, where samples were taken most regularly, densities seemed to decrease consistently during winter and spring, when food was presumably scarce and temperatures low (Figure S1).

Density was consistently lower in Le Perreux than in Franconville and we could not find any C. elegans in Le Perreux in April 2005 (Table 1). In addition, at most timepoints we could not find any C. elegans in Le Blanc and Hermanville compost heaps, even after processing copious amounts of compost sampled at different points in the heap (however, on several occasions we found Caenorhabditis briggsae, a relative of C. elegans).

Molecular diversity and outcrossing rate

A random subset of the isolated animals were genotyped at six microsatellite loci, named II-R, II-L, III-R, IV-L, V-L, X-R according to their chromosomal arm position (see Methods, HABER et al. 2005). Haplotypes are identified by the location name followed by a letter (Figures 4 andS2).

Molecular diversity: Diversity, as measured either by pairwise differences in microsatellite haplotype or by gene diversity over the six loci, showed striking variations between samples (Table 2). Whereas some samples (Obe-1005, Lis12-0705, all Fra, all HerC)

were very polymorphic, others (Per-0205, Per-0604, Per-0905, Mer1-0902, Mer2-0902, Mer3-0902, HerF-1105) were monomorphic. The low-density compost heap in Le Perreux was found to be less polymorphic than high-density compost heap in Franconville.

Heterozygote frequency and outcrossing rate: Out of the 540 individuals that we genotyped without prior selfing (see Methods), we found 10 heterozygotes, five of which were found in the Bla-1105 sample (Table 2) (this does not include the already described heterozygotes in samples Per-1004 andFra-1004;[START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF]. We calculated the equilibrium inbreeding coefficient f, an estimator of the inbreeding coefficient F is , from which we deduced the selfing rate s and outcrossing rate (1-s) (see Methods). Estimated outcrossing rates ranged from 0% to 7.6% (Bla-1105) (Table 2). Calculated over all diploid genotypes, the mean outcrossing rate of the different populations was 1.7% (c.i.: 1.1-2.5%).

This is very consistent with our previous results (1.3%) [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF]. In addition, the present results suggest that outcrossing rates vary between populations and provide an example of a population (Bla-1105) where outcrossing rates are significantly higher than in other samples, and closer to those found in SIVASUNDAR and HEY (2005) (ca 20%).

Male frequency and genotype:

We found two males out of 993 individuals (samples Fra-0805 and Obe-1005). Together with our previous samplings [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF], we obtained a total of four males for 2269 individuals, yielding an overall male frequency of 0.18%.

We placed each of the two males with unc-119 mutant hermaphrodites (bearing a recessive mutation resulting in uncoordinated movements) and scored non-Uncoordinated progeny. The male from the Fra-0805 sample sired no progeny, while the cross with the male from the Obe-1005 sample succeeded. Four of its progeny were genotyped at six loci and were found to be identical, indicating that the male was a homozygote at these six loci. The male may thus have been either a spontaneous male resulting from X chromosome nondisjunction, or a male resulting from biparental inbreeding (mating among identical genotypes).

Structure at different spatial scales

Within a single compost heap: In order to know whether the compost piles that we followed most closely (Franconville and Le Perreux) could each be considered homogenous, we measured spatial differentiation in genotype between samples from six different points within the pile (separated by 10 to 50 centimeters). In the three cases when enough individuals were isolated from the different samples and showed polymorphism, no significant genetic structure was found (Table 3). Thus, even though there were strong differences in density within a compost heap, we found no evidence of genetic structure within a heap. We therefore considered nematodes from a given compost pile to be genetically homogenous (what we call a population).

When comparing the genotypes of C. elegans individuals found in soil in Merlet or compost in Franconville, Primel and Sainte-Barbe, to those found on invertebrates (snails or isopods) sampled on the same habitat, no significant genotypic substructure was found either (Table 3).

Within a garden: structure at the scale of tens of meters: Strong and highly significant genetic structure at a given timepoint was found at the next spatial scale, when comparing samples within the same garden, in Hermanville (10-20 m), Merlet (10-100m),

Lisbon (100-300 m) and between the Primel and Sainte-Barbe compost heaps (1 km) (Table 3). This was in agreement with the very strong structure observed within the Merlet location in 2002 [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF].

Large-scale structure and haplotype sharing: At the global scale (100-1000 km scale), differentiation was significant between locations (Table 3), confirming our previous results (F st =0.78, BARRIÈRE and FÉLIX 2005). Except for two neighboring Primel and Sainte-Barbe locations (1 km apart), which share one haplotype at different timepoints (see below), only three instances of haplotype-sharing between locations were found (Bla-B = Her-A, Bla-N = Her-K, Bla-I = Mer-E).

In summary, C. elegans appeared to show no structure at a very small scale of a few centimeters, and very strong structure at scales above 10 meters; at a larger scale of hundreds of kilometers, structure appears weaker again, as previously observed [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF][START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF].

Temporal dynamics of genotypes

The main aim of our sampling was to analyze population dynamics over time. Genetic diversity fluctuated over time for a given locality, especially in Le Perreux, where it reached zero at several timepoints (Table 2). Most strikingly, allele frequencies varied dramatically, sometimes even between samples collected only six weeks apart. For the two localities followed most closely, Franconville and Le Perreux, we conducted pairwise comparisons of multilocus genotypes of consecutive samples. In Franconville, a single pair of consecutive samples was significantly differentiated, whereas in Le Perreux, several pairs were significantly differentiated (asterisks in Figures 2B and3B; Table 4).

In Franconville, where C. elegans could be isolated at all timepoints, the same major alleles at the three polymorphic loci (II-R, II-L and III-R) were conserved over time (Figure 2A). Allele frequencies fluctuated, but overall the population was rather stable.

By comparison, Le Perreux showed much stronger variations; after the density decreased in late winter 2004, and sample Per-0405 yielded no C. elegans, new alleles were found at loci II-R and X-R in June 2005 (Figure 3A). In July 2005, another replacement took place (at loci X-R and V-L). In September, the alleles and haplotypes found before April returned. Alleles found in June 2005 were found again in a single individual in December 2005 (Figure S2, haplotype Per-G). This strongly suggested an extinction of the population, followed by recolonization by new genotypes.

For the five other localities, we analyzed samples separated by one to three years.

Merlet 1 showed no significant differentiation over three years, the same major haplotype

Mer-E being present in both samples (Table 4, Figure S2D). Hermanville showed a marginally nonsignificant differentiation after Bonferroni correction, with a single minor haplotype (Her-D) being shared between the two timepoints (Table 4, Figure S2B). The three other localities (Le Blanc, Primel and Sainte-Barbe) showed very significant temporal differentiation (Table 4). Le Blanc witnessed an increase in genetic diversity that suggested an input from migration (several new alleles at several loci, Figure 4): the mutation rate was measured for locus II-L at 1.8 x 10 -4 and for locus IV-L at 2.7 x 10 -4 per generation [START_REF] Frisse | Understanding the mechanisms of microsatellite formation and mutation using the model organism Caenorhabditis elegans[END_REF] and these rates are too low to account for the increase in diversity observed in this locality. The Primel/Sainte-Barbe sampling locations (1 kilometer apart) each showed strong temporal differentiation. As noted earlier, the major haplotype (PriBar-B, Figure 4) of Pri-1004 was found at a high frequency in sample Bar-0805 while it was absent in samples Pri-0805 and Bar-1004, an indication that this temporal differentiation was in part due to migratory input.

Thus, natural populations of C. elegans can display dramatic changes in allele frequencies over short periods of time. In at least two localities (Le Perreux, Le Blanc), these variations were associated with density decline and subsequent recolonization events. In contrast, in two other locations with larger C. elegans populations (Franconville, Merlet 1) stable genotypes were maintained over several years.

Strikingly, alleles at different loci remained associated over time within a locality, suggesting little effective outcrossing (Figure 2B andS2). We therefore investigated the dynamics of linkage disequilibrium between loci.

Linkage disequilibrium

Multilocus linkage disequilibrium within a sample: The level of linkage disequilibrium between all loci was very high and significant for all polymorphic samples except Fra-1204 (Table S3). This is consistent with our previous observations based on AFLP data [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF].

Linkage disequilibrium over time:

A striking fact was the maintenance of very strong linkage disequilibrium between the same alleles for loci on different chromosomes over three years in Franconville. Indeed, the same two major multilocus genotypes in linkage disequilibrium were found throughout the three years, with very few recombinant genotypes (Figure 2B).

We looked for any evidence of decay of linkage disequilibrium in this locality. We calculated the linkage disequilibrium D between loci II-R and III-R, the two biallelic loci, and the associated D' (scaled by the maximum linkage disequilibrium level possible within the sample). We chose to work with the classical coefficient of linkage disequilibrium D because it is a simple statistic, whose decay equation is trivial (see below). Linkage disequilibrium levels were indeed very high, and remained high for more than three years (Figure 5). The sign of D' was the same in all samples, indicating that the polarity of linkage disequilibrium was conserved. The D' measure for the Fra-1204 sample was significantly lower than for later samples (comparing Fra-1204 with Fra-0106, the p-value=1.69 x 10 -5 ), which would indicate an increase (not a decrease!) in linkage disequilibrium over time. If we considered sample Fra-1204 as an anomaly and discarded it, the confidence intervals of D' for all samples were compatible. In any case, high linkage disequilibrium between the same alleles was maintained over three years. Given the density observed (see Table 1), the census size at the scale of the compost pile must be in the order of tens of thousands; therefore, drift alone cannot explain the absence of increase in recombinants frequency.

The outcrossing rate in Franconville was typical of results from all populations (0.9% over all timepoints), so the maintenance of linkage between alleles over such a long period of time was particularly puzzling. To test whether selfing alone could explain this level of linkage disequilibrium, we calculated the number of generations that would be compatible with the estimated outcrossing rate and the maintenance of linkage disequilibrium. The maximum decay of linkage disequilibrium compatible with our data would be from D' 0 =1 in generation 0 (upper bound of confidence interval for sample Fra-1102) down to D' N =0.947 in generation N (lower bound for sample Fra-0106). Using these numbers, we estimated (see Methods) the maximum number of generations N max as 6.0 generations over 38 months (one every 6.3 months). With the average outcrossing rate over all our samples, N max would be 4.2 generations (one every 9 months). These values are hardly compatible with the known generation time and life expectancy in C. elegans, which in standard laboratory conditions are 3.5 days and two weeks, respectively. The generation time could be much longer in the wild, depending on temperature, food availability, occurrence of diapause, etc. However, it appears unlikely that C. elegans reproduces with an average of two generations per year. Therefore, the maintenance of linkage between loci must be explained by other factors.

Selection against heterozygotes and recombinants? Outcrossing rates could be variable over time, for example seasonal or with a positive correlation between outcrossing and density. However, this cannot explain an increase in linkage disequilibrium. One possible alternative explanation would be partial reproductive isolation between the two haplotypes, either prezygotic (a lower rate of mating) or postzygotic (lower fitness of progeny from a cross between the two haplotypes). The former possibility was contradicted by the fact that recombinants between the two haplotypes were found, including some heterozygotes. To test the hypothesis of postzygotic isolation, we crossed two strains representing the two major haplotypes at the first timepoint (JU360 and JU361) and compared self-versus cross-progeny for progeny number and survival in the F2 generation. Embryonic lethality was high (4%) in the F2 progeny of F1 self-and cross-progeny; however, no significant difference was found between self-and cross-progeny in terms of brood size (means: 191.0 and 205.7, respectively, p-value: 0.42), embryonic lethality nor other obvious defects (Table S4).

DISCUSSION

The present results confirm previous findings on C. elegans genetic diversity, geographical structure and outcrossing rate based on heterozygote frequency. Most interestingly, they provide evidence for 1) highly dynamic populations, sometimes undergoing extinction and recolonization events, and 2) maintenance of linkage disequilibrium between loci over several years. We discuss the possible consequences of this dynamic aspect of C. elegans populations for its genetic and phenotypic evolution.

Spatio-temporal structure of C. elegans populations: metapopulation dynamics

The systematic sampling of several locations at different timepoints allowed us to obtain a dynamic picture of C. elegans natural populations. The metapopulation dynamics that we observed involve several temporal and spatial scales. The temporal scale of population turnover appears to be a few weeks or months, consistent with C. elegans being found on ephemeral habitats, such as fruits rotting below their tree or decaying snails, or at a longer timescale compost heaps and the surroundings of trees during the ripening period. The spatial scale of founding individual migration appears to be above one meter, with significant migration over very long distances, and no correlation between genotypic divergence and geographic distance [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF][START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF]). High selfing rates may increase apparent levels of structure by reducing the effective number of migrants [START_REF] Nordborg | Structured coalescent processes on different time scales[END_REF].

Different migration modes may operate at different spatial scales. Nematodes in the soil are able to move by themselves over a mean distance of 15 centimeters, and sometimes one meter, in a month [START_REF] Robinson | Nematode behavior and migrations through soil and host tissue[END_REF], which could explain the lack of structure at small scale. At a larger scale, migration may occur through vectors such as invertebrate associates (snails, isopods, etc.;[START_REF] Kiontke | Ecology of Caenorhabditis species in WormBook[END_REF]; the sample HerD-1105, recovered from a fly bait (see Table 1), supports this idea. Wind was described to be a potential long-distance (hundreds of meters to a few kilometers) migration vector for plant parasitic nematodes [START_REF] White | Wind-borne dispersal of potato-root eelworm[END_REF], and dust storms can be responsible for considerable movement of dauer larvae.

Indeed, dauers of C. elegans can survive dessication for several days at room temperature [START_REF] Kiontke | Ecology of Caenorhabditis species in WormBook[END_REF] and may use migration vectors that would appear unfit at first glance. Since we found a high density of C. elegans in rotting fruits, fruits may also be an efficient way to migrate -with the help of humans, flies or birds -over long distances.

Overall, the association of C. elegans with human-related habitats indicates that human activities could be responsible for large scale migration. The fact that C. elegans was seldom found in soil supports the idea of a patchy distribution of the species, with migration being a critical survival factor.

Outcrossing and maintenance of strong linkage disequilibrium

Three different measures provide information on the relative occurrence of selfing and outcrossing: heterozygote frequency, male frequency and linkage disequilibrium.

Heterozygote and male frequencies provide a measure at the short temporal scale of the previous generations. Our new estimates of heterozygote frequencies, yielding a global estimate of 1.7% outcrossing (c.i.: 1.1-2.5%), confirm our previous ones [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF]. It is improbable that many of these heterozygotes are due to mutational events, because half of them are heterozygotes at several loci (Dataset S1). If we consider selfing to be constant in the species, this 1.7% outcrossing rate predicts a male frequency of 0.85% (c.i.:

0.55-1.25%), not counting spontaneous males, whereas the observed male frequency is only 0.18% (95% c.i.: 0.05-0.45%). We cannot completely rule out that some males were missed in our sampling procedure, but the large discrepancy suggests that outcrossing rates vary over time and between different populations, as also suggested by the variation in outcrossing rate estimates among populations (Table 2).

Much more divergent is the 100-fold lower outcrossing rate estimate (10 -4 ) based on static estimates of linkage disequilibrium between loci in a local population [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF] or among worldwide isolates [START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF]. We find a very strong linkage disequilibrium between loci located on different chromosomes for most samples (Table S3).

CUTTER ( 2006) found similarly strong linkage on a worldwide scale (multilocus linkage disequilibrium I A S =0.29). Within a population of constant size, significant nonrandom associations between loci can appear by mutation and drift. In rapidly growing populations, like those undergoing metapopulation dynamics, such nonrandom associations are expected to be rare [START_REF] Slatkin | Linkage disequilibrium in growing and stable populations[END_REF]. Therefore, the observed linkage disequilibrium in C. elegans must have been present since the foundation of the population, possibly after colonization by two genotypes. Such a high level of linkage disequilibrium would then be expected to decay over time as a function of the outcrossing rate.

In our temporal surveys, we found no evidence of linkage disequilibrium decay over three years in the Franconville population. These observations are not compatible with the outcrossing rate estimated from heterozygote frequencies, unless the generation time is more than half a year. We found C. elegans mostly in the dauer stage, which could be responsible for a huge increase in generation time (in laboratory conditions, dauers may live up to eight months; C. Braendle, personal communication). However, the occurrence of only six generations in 38 months is improbable.

Several mechanisms can explain the discrepancy between the short-term outcrossing rate measured by heterozygote frequency and the maintenance of high linkage disequilibrium.

Population structure may explain high linkage disequilibrium among, but not within populations. The sampled populations could however be sink populations, receiving a constant flow of migrants from two populations, each monomorphic for one major haplotype: a Wahlund effect (WAHLUND 1928) could then explain the absence of decay of linkage disequilibrium at a given timepoint. Linkage should ultimately decay over time in this population, given that some sampled individuals appeared to be part of the reproductive pool (non-dauer stages). Our observations may however reflect stochastic effects associated with a small effective population size. Finally, an alternative hypothesis to explain our observations is selection acting against the cross-progeny (F1 or later generations). In laboratory conditions, we failed to find evidence of a strong effect on brood size of recombination of the two major Franconville haplotypes, yet it is possible that another character affecting fitness in natural conditions may be affected. Outbreeding depression was indeed observed between C.

elegans isolates, including within some of our local sets [START_REF] Dolgin | Inbreeding and outbreeding depression in Caenorhabditis nematodes[END_REF]. Thus, a possible explanation for maintenance of linkage disequilibrium is selection against heterozygotes or recombinants.

By comparison, in Drosophila melanogaster, linkage disequilibrium is very low and decays within 1 kb [START_REF] Long | Two sites in the Delta gene region contribute to naturally occurring variation in bristle number in Drosophila melanogaster[END_REF]. Even in a highly selfing species like Arabidopsis thaliana, linkage disequilibrium is lower than in C. elegans, at least at a large geographical scale: indeed, at this global scale, linkage disequilibrium is undetectable between different chromosomes and decays within approximately 50-250 kb for linked loci [START_REF] Nordborg | The extent of linkage disequilibrium in Arabidopsis thaliana[END_REF][START_REF] Nordborg | The pattern of polymorphism in Arabidopsis thaliana[END_REF]; linkage disequilibrium on a short region of 170 kb is I A S =0.179 [START_REF] Haubold | Recombination and Gene Conversion in a 170-kb Genomic Region of Arabidopsis thaliana[END_REF], weaker than on the complete genome of C. elegans, and unlike in C. elegans [START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF], some of it may be the result of spatial population structure [START_REF] Schmid | Evidence for a large-scale population structure in Arabidopsis thaliana from genomewid single nucleotide polymorphism markers[END_REF]. At a small scale within a patch of A. thaliana, linkage disequilibrium appears extensive, as in C.

elegans [START_REF] Bergelson | Genetic variation within and among populations of Arabidopsis thaliana[END_REF][START_REF] Nordborg | The extent of linkage disequilibrium in Arabidopsis thaliana[END_REF][START_REF] Stenoien | Genetic variability in natural populations of Arabidopsis thaliana in northern Europe[END_REF].

Possible consequences of C. elegans population dynamics on its phenotypic evolution

Both demographic and genetic results allow us to infer several consequences for the genetic and phenotypic evolution of C. elegans. The low outcrossing rate implies that alleles mostly occur in a homozygous state, hence purging of strongly deleterious recessive mutations should occur more readily than in outcrossing populations. Populations experiencing bottlenecks are likely to fix slightly deleterious mutations by genetic drift, but the strong reexpansion regimes that follow may allow compensatory mutations to occur.

Recent experimental evolution studies in C. elegans revealed that significant increase in fitness could already be seen after 10 generations [START_REF] Estes | Rapid fitness recovery in mutationally degraded lines of Caenorhabditis elegans[END_REF]. A transient loss of fitness, or of robustness of a given phenotypic character, followed by compensatory evolution may be a frequent mechanism of exploration of the genotype-phenotype landscape in C.

elegans. Furthermore, the almost exclusive selfing of C. elegans would be expected to favor co-evolution of its entire genome, thus resulting in outbreeding depression when outcrossing actually occurs [START_REF] Agrawal | Evolution of Sex: Why Do Organisms Shuffle Their Genotypes?[END_REF].

In addition, if the C. elegans metapopulation comprises source and sink populations, adaptation in the sink populations, which are doomed to extinction, is not relevant to future generations of the species as a whole, which would adapt only to source environments.

Identification of sink and source environments is thus crucial for the study of adaptive traits of C. elegans.

The very low overall genetic diversity of C. elegans [START_REF] Barrière | High local genetic diversity and low outcrossing rate in Caenorhabditis elegans natural populations[END_REF][START_REF] Haber | Evolutionary History of Caenorhabditis elegans Inferred from Microsatellites: Evidence for Spatial and Temporal Genetic Differentiation and the Occurrence of Outbreeding[END_REF][START_REF] Sivasundar | Sampling from natural populations with RNAi and population structure in Caenorhabditis elegans[END_REF][START_REF] Cutter | Nucleotide polymorphism and linkage disequilibrium in wild populations of the partial selfer Caenorhabditis elegans[END_REF]) cannot be explained by the mere two-fold reduction due to selfing; however, the observed metapopulation dynamics, in association with high rates of selfing, may result in selective sweeps that affect the whole genome and thus drastically reduce overall genetic and phenotypic diversity [START_REF] Charlesworth | Some evolutionary consequences of deleterious mutations[END_REF]. The population dynamics of C. elegans will thus affect molecular evolution patterns by reducing genetic diversity, increasing linkage disequilibrium, and potentially allowing the fixation of slightly deleterious mutations, which may then be , 1984, calculated with GDA;[START_REF] Lewis | Genetic Data Analysis: Computer program for the analysis of allelic data[END_REF] and its 95% confidence interval ('c.i.'); s: selfing rate calculated after f and its 95% confidence interval. [START_REF] Weir | Estimating F-statistics for the analysis of population structure[END_REF] with its confidence interval ('c.i.').

Figure 1 .

 1 Figure 1. Sampling locations.

Figure 2 .

 2 Figure 2. Temporal survey of allele and multilocus genotype frequencies in the

Figure 3 .

 3 Figure 3. Temporal survey of allele and multilocus genotype frequencies in the Le

Figure 4 .

 4 Figure 4. Multilocus genotypes in the Primel/Sainte-Barbe and Le Blanc populations at

Figure 5 .

 5 Figure 5. Evolution of linkage disequilibrium over time in the Franconville population.

F

Table 1 . Sampling of C. elegans populations.

 1

	Summary description of C. elegans population sampling. 'Location': name of sampling
	location; 'Sample': name of sample; 'Date': date of sampling; 'Mass': mass of sample (in
	grams); 'Ind': number of C. elegans individuals recovered, 'Dens': density of sample
	(individuals per gram); 'Nature': nature of sample and habitat; 'Stages/remarks': occurrence
	of different developmental stages when we were able to determine it, and remarks about the
	sample. The four larval stages are indicated as L1-L4. 'd': dauer. 'ad': adult. See Table S1 for
	more details, especially of subsamples.

Table 2 . Molecular diversity and outcrossing rates in the sampled C. elegans populations.

 2

	'Inbred?': whether individual genotypes were scored after laboratory inbreeding ('i', inbred
	strains) or without ('h', potentially heterozygous). 'N': number of genotyped individuals.
	'Pair.diff.': mean number of different pairwise microsatellite loci; 'H': gene diversity
	(calculated with Arlequin; EXCOFFIER et al. 2005) and its standard deviation ('s.d.'); 'Θ hom ':
	theta parameter calculated after homozygosity and its standard deviation; 'Het': number of
	heterozygotes; 'Ho': observed heterozygosity; f: equilibrium inbreeding coefficient (estimator
	of F is after WEIR and COCKERHAM

Table 3 . Spatial structure of genetic differentiation at different scales.

 3 'p-value': p-value of differentiation test after Bonferroni correction. Values below 0.05 indicate a significant spatial structure. 'θ': estimator of F st after WEIR and COCKERHAM(1984) with its confidence interval ('c.i.').

	Sample	p-value	θ	c.i.
	Structure within one compost pile
	Fra-0106	0.990
	Fra-0905	0.650
	Per-1205	0.792
	Mer1-1005 (snails-compost)	0.792
	Pri-0805 (snails-compost)	0.792
	Fra-1004(snails-compost)	0.133
	Bar-0805 (isopods and compost)	0.420
	Within one garden
	Lisbon	<0.001	0.366 0.189-0.504
	Hermanville	<0.001	0.255 0.207-0.295
	Merlet 05	<0.001	0.776 0.598-0.886
	Primel-Sainte-Barbe 04	<0.001	0.591 0.453-0.664
	Primel-Sainte-Barbe 05	<0.001	0.443 0.270-0.670
	Merlet 02	<0.001	1.000 1.000-1.000
	Large scale
	All locations (Jul-Nov 05)	<0.001	0.532 0.445-0.605

Table 4 . Temporal structure of genetic differentiation.

 4

'p-value': p-value of differentiation test after Bonferroni correction; 'θ': estimator of F st after

Acknowledgments

We are very grateful to our colleagues and frequent compost providers, J.-A. Lepesant and C. Pieau, and to all those who maintained the gardens that were sampled. We thank D. Higuet and B. Toupance for help and advice with the analyses, and C. Braendle, D. Charlesworth, A. Cutter, E. Dolgin as well as two reviewers for helpful comments on the supported by the Ministry of Research of France and the Association pour la Recherche sur le Cancer. This work was supported by the Centre National de la Recherche Scientifique and the Ministry of Research of France through a Biological Resource Center grant.

In some cases, several samples were taken at the same date; the density indicated is the mean, and standard error is indicated by an error bar.


G Vasile

E Trouv É

L Valet

J.-M Nicolas
email: nicolas@tsi.enst.fr

L Bombrun

M Gay

I Petillot

Ph Bolon

V Buzuloiu
email: buzuloiu@alpha.imag.pub.ro

Coherent-stable scatterers detection in SAR multi-interferograms: feature fuzzy fusion in Alpine glacier geophysical context

SAR interferometry (InSAR) performs two acquisitions (spatially separated by the baseline) of the signal by the resolution cell which contains height and/or displacement information. Repeat pass spaceborne interferometry provides multi-interferograms which can be used to extract such information either by combining the multi-temporal results of conventional interferometry or by a different approach based on specific targets: the coherent stable scatterers (CSS). In this paper a two-step approach is proposed to obtain specific features from multi-temporal InSAR data sets. The first step consists in extracting image attributes related to the useful information. The second step consists in merging the attributes using an interactive fuzzy fusion technique. The interactive fuzzy fusion is proposed to provide end-users with a simple and easily understandable tool for tuning the detection results. The method is applied on a data set of five co-registered ERS 1/2 tandems from the French Alps (the Mont-Blanc region), including two temperate glaciers: the Argentière and the Mer-de-glace. The results illustrate how the end-user can combine the proposed attributes to detect the presence of CSS or distributed stable scatterers usefull for multi-temporal analysis.

I. INTRODUCTION

Synthetic Aperture Radar interferometry (InSAR) has the potential to measure temperate glacier displacement with a large coverage compared to sparse terrestrial ground measurements [START_REF] Trouvé | Combining airborne photographs and spaceborne SAR data to monitor temperate glaciers. potentials and limits[END_REF]. Depending on the surface cover (rocks, snow or ice), the scattering properties may change with time and induce the decrease of interferogram coherence i.e. temporal decorrelation. More recent techniques have been proposed to select in an interferogram the pixels which exhibit less decorrelation than others in order to reduce such effects within conventional InSAR processing. Several discriminant parameters are used to describe these coherent or stable scatterers: either within one SAR acquisition (internal coherence [START_REF] Souyris | On the use of complex SAR image spectral analysis for target detection: Assessment of polarimetry[END_REF], mutual information) or in a large number of interferograms [START_REF] Hooper | A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers[END_REF].

In this paper a two-step approach is proposed to obtain simultaneously spatial and temporal information from multi-temporal InSAR data sets. The first step consists in extracting image attributes related to the useful information. The second step consists in merging the attributes using a interactive fuzzy fusion technique. The interactive fuzzy fusion is proposed to provide end-users with a simple and easily understandable tool for tuning the detection results. The interactive aspect allows the integration of prior information, expert knowledge, and/or new available data. The symbolic aspect is about the type of effective fusion of the data: in opposition of the numerical way where the values are directly merged by numerical operators, in a symbolic way a semantic description of the data is made and the fusion is realized at the symbol level which is close to the end user reasoning.

The proposed method is illustrated on a data set of five corregistered ERS 1/2 InSAR tandems from the French Alps (the Mont-Blanc region), including two temperate glaciers: the Argentière and the Mer-de-glace. The estimated attributes are:

• a new attribute to characterise the interferometric coherence temporal stability, the "multi-InSAR coherence", which combines the multi-temporal coherences estimated by IDAN technique [START_REF] Vasile | Intensity-drivenadaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation[END_REF]. • an attribute which characterizes the target coherence in each interferogram, the "sub-band coherence" [START_REF] Souyris | On the use of complex SAR image spectral analysis for target detection: Assessment of polarimetry[END_REF],

• a texture multi-temporal feature derived from the second kind statistics (the 3rd order log-cumulants) which characterize the presence of coherent targets from their intensity in the time series [START_REF] Bujor | Application of log-cumulants to the detection of spatiotemporal discontinuities in multitemporal SAR images[END_REF], The fuzzy fusion of such attributes provides information on the preservation of the interferometric coherence over the studied glaciers for velocity measurements, both on pointwise and distributed targets, usually called coherent or non-coherent targets according to the length of the antenna aperture where the target back-scattered signal remains coherent. Moreover, possible candidates for coherent and stable scatterers (CSS) are obtained for further CSS processing [START_REF] Hooper | A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers[END_REF].

II. ATTRIBUTE ESTIMATION

The first step of the proposed approach consists in developing attributes which reveal the presence of specific features in the repeat pass multi-temporal series of interferometric SAR images.

A. Multi-InSAR coherence

In multi-InSAR processing, it is important to have a good description of both scattering properties and temporal evolution of the target area, over the whole dataset.

One parameter for characterizing the stable scatterers within multiple interferograms is proposed, namely the "multi-InSAR coherence" C v . This parameter is estimated using intensity-driven Adaptive-Neighborhood AN [START_REF] Vasile | Intensity-drivenadaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation[END_REF] technique:

C v = N k=1 i∈AN z M i k z S * i k N k=1 i∈AN |z M i k | 2 • N k=1 i∈AN |z S i k | 2 , ( 1
)
where N is the number of available InSAR couples, z M and z S are the master and slave SLC images, respectively. In Eq. ( 1), it is important to notice that the local deterministic component of the phase signal φ should be flattened when estimating the complex correlation within the 3D adaptive neighborhood. In this paper the adaptive HR-LR model of local frequencies (f x , f y ) [START_REF] Vasile | High resolution SAR interferometry: estimation of local topography in the context of glacier monitoring[END_REF] is used for the phase flattening:

φ f l (k, l) = φ(k, l)-2π[(k -m)f x +(l -n)f y ]-φ(m, n), (2)
where (m, n) is the position of the central pixel within the interferogram and (k, l) ∈ AN belong to the estimation sliding window.

B. Sub-band coherence

One parameter for characterizing the coherent scatterers within one interferogram is the "sub-band InSAR coherence" C sub . Firstly, the interferometric pair is decomposed in B sub-bands as in [START_REF] Souyris | On the use of complex SAR image spectral analysis for target detection: Assessment of polarimetry[END_REF]. Then, this 3D coherence is estimated within a spatial neighborhood W as follows:

C sub = B j=1 i∈W s M ij s S * ij B j=1 i∈W |s M ij | 2 • B j=1 i∈W |s S ij | 2 , (3)
where s M and s S are the master and slave sub-bands, respectively.

C. 3rd Order Log-cumulant

A different way to detect the strong intensities of CSS in multi-temporal data consists in estimating texture parameters in spatio-temporal windows made of the pixel of a small spatial neighborhood at the different dates.

The second kind statistics are well suited to estimate the parameters of the probability density functions defined on R + , such as the Gamma distributions for speckle intensity images. The second and third order log-cumulants have been proposed to reveal spatial features and temporal changes [START_REF] Bujor | Application of log-cumulants to the detection of spatiotemporal discontinuities in multitemporal SAR images[END_REF]. In homogeneous areas, the third order logcumulant κ3 is close to the theoretical value Ψ ′′ (L) where Ψ ′′ is the second derivative of the Digamma function and L the number of looks. In the presence of heterogeneous features such as strong intensity CSS, κ3 values increase because of the heavy tail of the intensity distribution. It provides a complementary attribute to reaveal potential CSS from the multi-temporal intensity information.

III. INTERACTIVE FUZZY FUSION SYSTEM

The second step of the proposed approach combines the information extracted from the SAR images. Expert knowledge is introduced by using an interactive symbolic fuzzy fusion (ISFF) method [START_REF] Valet | A fuzzy linguisticbased software tool for seismic image interpretation[END_REF]. The interactive and symbolic aspects are the main keys of the method detailed hereafter.

A. Knowledge representation

The attributes to be fused are described by the experts by means of several symbols (words). These symbols are related to the attribute numerical values through the membership functions. For each attribute

A i , i ∈ [1, N A ],
where N A is the number of attributes, a symbolic description is a set of s i words denoted by:

L i = {L i,1 , L i,2 , ..., L i,si }
The membership functions are set by the system with equidistant modal values, but they can be adjusted later by the user (Section III-C).

The attributes represented by sets of symbols are aggregated according to the expert knowledge. This knowledge is modeled using the fuzzy subset theory, the experts being able to express their reasoning in a linguistic way by means of IF-THEN rules. A general rule example for the description of an output class is:

IF A 1 is L 1,j AND A 2 is L 2,j ′ AND ... THEN pixel BELONGS TO out k .
where

A i , i ∈ [1, N A ] are the attributes, L i,j , j ∈ [1, s i ]
the symbols that describe these attributes, and

out k is an output class, k ∈ [1, N C ],
where N C is the number of classes.

The fuzzy subset theory is an adequate tool to encode such linguistic knowledge [START_REF] Dubois | Performing approximate reasoning with words -In Computing with words in Information/Intelligent systems 1[END_REF]. In this theory, a rule-base is used to represent all the rules. In a three-input fusion case, this rule-base can be described in the three-dimensional (3D) attribute domain where each sub-cube represents a rule for a desired output class.

B. Fusion principle/aggregation step

The numerical input attributes are processed according to the membership functions in order to obtain a linguistic description. The aggregation of the linguistic descriptions is then performed by Zadeh's compositional rule of inference in order to obtain at each pixel the membership degree µ k to the output class k:

µ k (a 1 , a 2 , ...) = ⊥ L1j ,L 2j ′ ,...∈L1,L2,... ⊤ ⊤(µ D(a1) (L 1j ), µ D(a2) (L 2j ′ ), ...), µ R (L 1j , L 2j ′ , ...k)
where (a 1 , ...) are the values taken by the attributes A 1 , ... at the current pixel. The term µ R (L 1j , ..., k) is equal to 1 if the corresponding rule is present in the rule-base and 0 otherwise. The fuzzy operators involved in this formula are chosen to obtain a system behavior close to the behavior of experts when they interpret images [START_REF] Valet | A fuzzy linguisticbased software tool for seismic image interpretation[END_REF]. The selected operators are:

• ⊤(a, b) = a * b • ⊥(a, b) = min(a + b, 1)
Finally, the decision step consists in classifying the considered image pixels into output classes according to the greatest output membership degree µ k .

C. Graphic user interface

The proposed approach involves experts in the fusion of the attributes. The experts' role consists in initializing the fusion system thanks to their experience. They give the rules which define the sought-after classes and they provide the membership functions in order to adjust the meanings of the words used in the rules. These parameters must be set in an interactive way, in order to allow them to improve the detection. The other steps of the fuzzy method (the inference rule and the combination/projection operator) are automatic: the end users are not expected to interact in these steps which require specific knowledge in the fuzzy subset theory.

A Graphic User Interface (GUI) has been developed to enable an interactive introduction of the expert knowledge into the fusion of three attributes. The GUI is composed of a 3D space where each axis corresponds to an attribute. The linguistic parameters and the rules are easily set by experts directly in the representation cube. For each rule, the corresponding sub-cube can be stretched and the membership functions which describe the symbols involved in the rule are automatically adjusted in the same 3D space. In order to help experts in this adjustment, different sets of points corresponding to reference regions (pointed out by experts or given by masks), can be drawn in the attribute space. The objective is then to stretch the sub-cubes in order to isolate the different sets of points as precisely as possible. Then the system is ready to perform the fusion. The fuzzy results can be observed as images of the membership degrees to the output classes. Crisp results are obtained either interactively by thresholding one of the membership degrees or automatically by selecting the output class which obtain the highest degree.

IV. RESULTS AND DISCUSSION

The proposed method has been developed to deal with the high mountainous relief present in the Chamonix Mont-Blanc test site which is located in the Alps, near the borders between France, Italy and Switzerland (45 • 50 ′ N, 6 • 51 ′ E). It includes the Aiguille-Verte (4122 m ASL), the Chamonix valley (1000 m ASL) and two instrumented glaciers: the Mer-de-Glace and the Argentière.

In this paper, 10 SAR images from ERS 1 and ERS 2 have been processed. This data set presented in Tab. I corresponds to 1-day tandem couples acquired in descending passes. After obtaining the SLC images, the five tandem interferograms were produced with an initial 2 × 10 complex multi-looking. Fig. 1 presents a color composition of three SAR amplitudes and coherences at three different months: October, January and March. One can notice that in Fig. 1, the gray regions correspond to non-changing features, while the colored regions exhibit an important change in radiometry.

After coarse corregistration of the five interferograms, the third order log-cumulant κ3 of the SAR amplitudes is computed within a 3 × 3 × 10 window (Fig. 2-(a)). This parameter provides information on the stability of strong scatterers. For the sub-band coherence attribute, each SLC image is decomposed into 5 sub-bands with 10% overlapping and Hamming weighting. The resulting sub-band coherences C sub are obtained according to Eq. 3 with a 3 × 3 boxcar window. Then, the sub-band coherences obtained for the different InSAR couples are combined (Fig. 2-(b)): as for the CSS detection the objective is to have an important coherence level in all coherence maps, the aggregation is performed by using the minimum operator:

C min sub = min j∈{1,..,B} C subj . (4)
For the multi-InSAR coherence attribute, each 20-look interferogram is filtered independently by using IDAN technique [START_REF] Vasile | Intensity-drivenadaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation[END_REF] and compensating the deterministic phase component [START_REF] Vasile | High resolution SAR interferometry: estimation of local topography in the context of glacier monitoring[END_REF]. The multi-InSAR coherence illustrated in Fig. 2-(c) is computed according to Eq. 1.

Once the space defined by the three attributes from Fig. 2 is built, the 3D histogram is computed Fig. 3-(a). As the third order log-cumulant provides information on the strong scatterers within the SAR amplitude images, its distribution presents a main lobe in the vicinity of κ3 = Ψ ′′ (20) and outliers corresponding to strong scatterers. This induces a non-uniform distribution of the words employed in the fuzzy fusion step. After visualizing the 3D histogram, the end-user must tune the partition of the attribute space, assuring enough word sampling in each cloud of points.

The final step, before starting the fusion is to assign the set of rules corresponding to the desired classes. This can be performed by:

• expert (a priori) knowledge on the behaviour of the attributes. For example, if the sub-band coherence, the multi-InSAR coherence and the third order logcumulants are large, then the output class is CSS (rules illustrated in red in Fig. 3-(b)); • visualizing clouds of points corresponding to different region of interests which are candidates for the desired classes. This regions of interest can be selected on either one of the attributes or on another image easier to interpret (ex: one of the SAR amplitude images).

The detection result is presented in Fig. 2-(d). The class marked in red represents the CSS points. As the obtained CSS points are only on non-moving areas (mainly in the Chamonix valley), CSSInSAR processing [START_REF] Hooper | A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers[END_REF] for measuring glacier velocity is not applicable with ERS data on the studied glaciers. This behaviour is confirmed also over the distributed targets: only the rocky slopes are detected as they are not affected by the seasonal temperature variations. Since the ice/snow areas are rejected, the PSInSAR [START_REF] Ferretti | Permanent scatterers in SAR interferometry[END_REF] analysis for glacier velocity measurements is also not applicable with ERS tandem data. Accordingly, both alternative techniques to conventional InSAR can not be used with annual interferogram stacking covering the summer season.

V. CONCLUSION In this paper, a tunable fuzzy fusion system is proposed for the detection of specific features from multitemporal InSAR data sets: either distibuted stable scatterers with sufficient interferometric coherence for PSInSAR measurements, or CSSInSAR for displacement measurement based on specific targets. The nature of the application requires interaction between the intrinsic properties of SAR scatterers and qualitative information brought by experts. The proposed fusion system is designed to combine different sources of information by means of a fuzzy symbolic set of rules, the behavior of which is close to that of the end-user. The example presented on the Mont-Blanc temperate glaciers shows encouraging results in combining qualitative geophysical knowledge with objective parameters for detection-classification purposes: coherent stable scatterers and distributed stable scatterers selection. In perspective, it will be interesting to use such approaches in the analysis of future POL-InSAR data acquired over Alpine glaciers.

Fig. 1 .

 1 Fig. 1. Chamonix-Mont Blanc [512×512 pixels]: (a) color composition of IDAN filtered SAR amplitudes (Red=1996/03/11, Green=1996/01/01, Blue=1995/10/23), (b) color composition of IDAN estimated SAR coherences (Red=1996/03/11, Green=1996/01/01, Blue=1995/10/23).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Multi-InSAR attributes and fuzzy fusion results [512 × 512 pixels]: (a) κ3 third order log-cumulant (3 × 3 × 5 boxcar), (b) C min sub minimum of the five coherences (3 × 3 boxcar), (c) Cv IDAN multi-InSAR coherence, (d) detection result (Red=CSS and Blue= distributed stable scatterers).

TABLE I ERS

 I SAR RAW DATA ACQUIRED OVER MONT-BLANC AREA WITH THE FRAME NUMBER 2673 AND B: BASELINES (PARALLEL AND PERPENDICULAR TO THE RADAR LINE OF SIGHT).

	Dates	Orbites	B (m) B ⊥ (m)
	95.07.09/95.07.10 20823/01150	-4	29
	95.10.22/95.10.23 22326/02653	-44	-107
	95.12.31/96.01.01 23328/03655	79	208
	96.03.10/96.03.11 24330/04657	26	9
	96.04.14/96.04.15 24831/05158	39	93
	(a)		(b)

ACKNOWLEDGMENTS

This work was supported by the French national project ACI-MEGATOR. The authors wish to thank to the European Space Agency for providing ERS SAR data through the Category 1 proposal No.3525.