Julien Cucherousseta
email: julien.cucherousset@cict.fro.cucherousset

Frédéric Santoula
email: santoul@cict.fr

Ordi Figuerolab

Régis Céréghinoa

EcoLab Régis Céréghino
email: cereghin@cict.fr

How do biodiversity patterns of river animais emerge from the distributions of common and rare species?

Keywords:

Introduction

ABSTRACT

We studied the patterns of commonness and rarity for one vertebrate (fish) and four freshwater insect taxa (Ephemeroptera, Plecoptera, Trichoptera and Coleoptera) in southwestern France (57,000 ォ ュ セ L @ and we analysed the relationships between the location of sites and the contribution of commonness and rarity to species richness within a large stream system. Richness patterns in fish and aquatic insects were related to the location of sites within the stream system. The number of common and rare fish species increased from up-to downstream areas as a result of downstream additions of species. The number of common insect species peaked in the intermediate section of the river continuum, whereas rarity increased with decreasing elevation. In all taxa, common species gave a doser approximation to overall patterns of species richness than did rare ones. The biodiversity patterns of river animais emerged from convergence in the distributions of common and rare species (fish), or mostly from the distribution of common species (insects). However, in fish, Ephemeroptera and Plecoptera, the rarer species became almost equally, or more strongly correlated with overall species richness when increasing information along the common-to-rare and rare-to-common sequences. These patterns suggested that rarer species show a similar or stronger affinity, on a species-for-species basis, for high richness areas than do the commoner species. These schemes have implications for biodiversity assessments, as studies using common species richness to target important areas for monitoring or conservation efforts within stream systems will not necessarily identify areas important for rare species, and vice versa.

Understanding how biodiversity patterns emerge from the distributions of rare and common species is a key concern of conservation biology [START_REF] Murray | How plant life-history and ecological traits relate ta species rarity and commonness at varying spatial scales[END_REF][START_REF] Storch | Patterns of commonness and rarity in central European birds: reliability of the coresatellite hypothesis within a large scale[END_REF][START_REF] Lennon | Contribution of rarity and commonness ta patterns of species richness[END_REF][START_REF] Pearman | Common species determine richness patterns in biodiversity indicator taxa[END_REF]. On one hand, rare species are regarded as having a high conservation priority {e.g., Prendergast et al., 1993;[START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF], because local rarity may increase the likelihood that demographie and/or environmental stochasticity will eliminate populations. Indeed, a restricted spatial distribution (with individuals occurring with high or low densities) implies that populations will probably experience adverse conditions simultaneously [START_REF] Gaston | Rarity as double jeopardy[END_REF]. On the other hand, our understanding of the determinants of overall patterns of species richness may gain most from consideration of why common species occur in sorne areas and are absent from others, rather than from consideration of the distributions of rare species {Vazquez and Gaston, 2004). Thus, even if rarity and commonness cannot be always explained, patterning their geographie variations and understanding their respective contribution to biodiversity patterns remains a very important part of conservation biology [START_REF] Chu | Comparative regional assessment of factors impacting freshwater fish biodiversity in Canada[END_REF][START_REF] Devictor | Can common species benefit from protected areas?[END_REF].

Most of our current understanding of commonness and rarity has come from studies ofterrestrial plants, birds, mammals and sorne insects (e.g., [START_REF] Thomas | Rarity, species richness and conservation: butterflies of the Atlas Mountains in Morrocco[END_REF][START_REF] Berg | Common and rare Swedish vertebrates-distribution and habitat preferences[END_REF][START_REF] Lennon | Contribution of rarity and commonness ta patterns of species richness[END_REF]Vazquez and Gaston, 2004). Studies on commonness and rarity in freshwater ecosystems have mainly been conducted from a monitoring and assessment point of view [START_REF] Cao | Rare species in multivariate analysis for bioassessment: sorne considerations[END_REF]. Indeed, most bioassessment systems use common species to assess the ecological quality of surface waters [START_REF] Nijboer | The myth of communities determining ecological quality of surface waters using macroinvertebrate community patterns[END_REF], while rarity can be used in assessing the conservation value of freshwaters [START_REF] Nijboer | Rare and common macroinvertebrates: definition of distribution classes and their boundaries[END_REF]. Other works addressed biodiversity aspects such as the loss of information induced by omitting rare species in ecological studies [START_REF] Cao | How important are rare species in aquatic community ecology and bioassessment?[END_REF][START_REF] Marchant | How important are rare species in aquatic community ecology and bioassessment? A comment on the conclusions of Cao et al[END_REF][START_REF] Arscott | Role of rarity and taxonomie resolution in a regional and spatial analysis of stream macroinvertebrates[END_REF]. However, freshwater habitats remain poorly represented in published studies on commonness and rarity, when compared to other systems {Chapman, 1999, but see [START_REF] Faith | Correlation of environmental variables with patterns of distribution and abundance of common and rare freshwater macroinvertebrates[END_REF][START_REF] Malmqvist | Threats ta the running water ecosystems of the world[END_REF][START_REF] Dudgeon | Freshwater biodiversity: importance, threats, status and conservation challenges[END_REF]. At the same time, available information suggests that freshwater biodiversity has declined as a whole faster than either terrestrial or marine biodiversity over the past 30 years Oenkins, 2003). Whatever the studied organisms, considering rarity and commonness through numbers of species rather than in terms of species assemblages sensu stricto is likely to fit with a broader modelling approach, because the resulting patterns are not expected to be region-specifie (i.e., any model only referring to a region-specifie list of species is more prone to have local acceptance). Subsequently, modelling patterns ofbiodiversity such as commonness and/or rarity means that models will predict species richness by using environmental conditions (e.g., elevation, temperature) as explanatory variables {Park et al., 2003). With a large number of variables, it is possible to "overfit" the models so that they may perform well in the context of the dataset used to generate them, but fail to be robust when used elsewhere {Rushton et al., 2004). Therefore, the use of species richness and of a limited number of explanatory variables in a successful final model would be relevant to derive broadly acceptable schemes, but could also reduce the effort and cost of data collection for environmental management applications {Céréghino et al., 2005).

Assuming that common species have wider fundamental niches (wide range) than rare species {Rosenweig and Lomolino, 1997), we hypothesized that common and rare species would differ in their responses to river habitat conditions, whatever the study taxa. Subsequently, because of differences in the ecological requirements and biological traits, we predicted that rarity and commonness would make different contributions to the overall patterns of species richness depending on the considered taxa. We thus considered the distributions ofboth common and rare species to understand how rarity and commonness contribute to the overall biodi-versity patterns in different freshwater taxa. We focused on both vertebrate (fish) and invertebrates (four insect orders, namely Ephemeroptera, Plecoptera, Trichoptera and Coleoptera), these five groups being commonly considered at the species level in basic and applied freshwater studies, and being particularly sensitive to the impact of human activities and changes in ecosystem features {Resh and Jackson, 1993; [START_REF] Oberdorff | Is assemblage variability related to environmental variability? An answer for riverine fish[END_REF][START_REF] Park | Applications of artificial neural networks for patterning and predicting aquatic insect species richness in running waters[END_REF]. Moreover, freshwater insects are an important source of nutrition for fish, and this strong trophic link provides further justification for studying these groups together. Specifically, we sought: (i) to produce explicit models which allow to understand the relationships between the location of sites within the stream system and commonness, and (ii) to analyse the contribution of rarity and commonness to patterns of aquatic species richness. The results were used to draw a conceptual diagram summarizing patterns in species richness (rare, cornmon and overall) within large river catchments. These schemes will have implications for biodiversity assessments, because studies using common species richness to target important areas for monitoring or conservation efforts within stream systems will not necessarily identify areas important for rare species, and uice uersa.

Methods

2.1.

Study area and data collection

The River Garonne {525 km long) has its source in the Maladetta Glacier {Spain), and reaches the Atlantic Ocean through the Gironde estuary {France; Fig. 1). The River Garonne stream system drains an area of about 57,000 km 2 , with a mean annual discharge of about 545m 3 s-1 . Compared with other French rivers (e.g., River Seine and River Rhône), the River Garonne is less disturbed by industrial pollution [START_REF] Oberdorff | Development and validation of a fish-based index (FBI) for the assessment of "river health" in France[END_REF]. The elima te of the region is influenced by oceanic processes, but this declines to the southeast where it undergoes the Mediterranean influence with dry winds and weaker rainfalls. Fish and insects were collected at unstressed sites. These sites where evenly distributed throughout the Garonne stream system {Fig. 1), and were not subjected to anthropogenic impacts such as flow regulation, chemical pollution, or urban runoff (indexed by the French Water Agency: http://www.eau-adour-garonne.fr/; see also [START_REF] Park | Applications of artificial neural networks for patterning and predicting aquatic insect species richness in running waters[END_REF]. The stream length selected for monitoring at each site was about ten times the stream width {from 50 rn in headwaters to 500 rn in plain areas).

Five hundred and fifty-four sites ranging from high mountain {2500 rn a.s.l.) to plain or coastal {10 rn a.s.l.) areas were sampled for fish. Fish data were collected between 1980 and 2000. All sites were sampled once by electrofishing, during low-flow periods, using two-pass removal sampling. Thirtynine fish species were identified (detailed list in Santoul et al., 2004) among which 24 were native species. In this study, exotic fish {15 species) were not considered when selecting common species to allow comparisons with previous studies on rarity (i.e., Santoul et al., 2005). In the River Garonne, most fish where widespread, whereas few species had a local distribution. Following Santoul et al. {2005) , we selected "common species" as those species occurring in more than 5% of the sampling sites, i.e., having a wide range sensu [START_REF] Lennon | Contribution of rarity and commonness ta patterns of species richness[END_REF].

One hundred and fi.fty-five sites ranging from 10 to 2500 rn a.s.l. were sampled for aquatic insects. Samples were taken from 1988 to 1998 and each site was sampled twice (swnmer and winter). At each period, eight sample-units were taken from all substratum types using a standard Surber sampler (sampling area 0.1 m 2 , mesh size 0.3 mm). Sampleunits were distributed in proportion to the relative abundance of the various substrata (e.g., gravels, pebbles, vegetation, etc.). This scheme was intended to ensure that the substratum types sampled at the various sites reflected the withinsite and the between-sites diversity of habitat conditions, i.e. the continuous gradient of physical characteristics of streams from headwaters to plains. Note that the fi.sh and insect samples were taken at different sites, to avoid disturbing the benthos (notably during the summer sampling sessions). Ail samples were fully sorted in the laboratory. Ephemeroptera, Plecoptera, Trichoptera and Coleoptera (hereafter called EPTC} were identified to species. 'IW"o hundred eighty-three EPTC species were identified (detailed list in [START_REF] Céréghino | Spatial analysis of stream invertebrates distribution in the Adour-Garonne drainage basin {France), using Kohonen self organizing maps[END_REF][START_REF] Céréghino | Using self-organizing maps ta investigate spatial patterns of nonnative species[END_REF], with bath restricted range (i.e., local distribution} and widespread species (i.e., longitudinal or regional distribution). As for fish, we used previous inventories [START_REF] Céréghino | Spatial analysis of stream invertebrates distribution in the Adour-Garonne drainage basin {France), using Kohonen self organizing maps[END_REF][START_REF] Céréghino | Using self-organizing maps ta investigate spatial patterns of nonnative species[END_REF] to select "common species" as those species occurring in more than 5% of the sampling sites.

2.2.

Data analyses

To analyse the relationships between the location of sites within the stream system and commonness, each sampling site (for fi.sh or insects) was characterised with four environmental variables: elevation (a.s.l., m}, distance from the source (km), stream order, and reach slope (r.J. The implicit use of spatial location in building regression models to explain biodiversity patterns was specifically intended to keep models simple and generally applicable. The dependent variable used (number of common species) corresponds to count data and a general linear modelling (GLM) was used because it allows a more versatile analysis of correlation than standard regression methods. For analysing common species we fi.tted models using a Poisson distribution and a log link function [START_REF] Crawley | Glim for Ecologists[END_REF]. River was included as a mndom factor in the model to control pseudoreplication due to the inclusion of more than one point from each river. 'lb correct the possible effects of under-or over-dispersion on statistical tests, deviances were scaled with the square root of the ratio deviance/degree of freedom (SAS Institute, 2000). Data were analysed with the GLIMMIX macro for SAS 8.2 {SAS Institute, 2000), fitting a mixed effects general linear mode] (river as a random variable and environmental variables as fixed variables}. Main effects were fitted using type III tests and a stepwise backwards removal procedure was used to obtain a final model containing only signifi.cant factors.

To further analyse the contribution of commonness and rarity to patterns of aquatic species richness in rivers, we ranked all species in each species group from the most widespread to the most restricted (common-to-rare), and from the most restricted to the most widespread (rare-to-common), and generated a sequence of patterns of species richness for increasing numbers of species (see details in [START_REF] Lennon | Contribution of rarity and commonness ta patterns of species richness[END_REF]Vazquez and Gaston, 2004). At each stage along bath series of richness patterns, we correlated the species richness pattern for the sub-assemblage with the full assemblage. This allows us to compare sub-assemblages of the n most common with the n most rare species, in terms of how weil they match the full assemblage richness pattern. We also plotted, along the common-to-rare and rare-to-common sequences, the sequential richness pattern correlations against a simple measure of accwnulated information in species distributions, the cumulative sum across species of the binomial variance EJ';(1 -pJ, where Pi= proportion of study area occupied by the ith species (i.e. number of sites occupied divided per the nwnber of sites sampled, as per [START_REF] Lennon | Contribution of rarity and commonness ta patterns of species richness[END_REF]Vazquez and Gaston, 2004}. Finally, we swnmarized the results on the distributions of common species and the contribution of rarity and commonness with those from Santoul et al. ( 2005) on the distributions of rare species in a more readily interpretable manner, i.e., a schernatic diagram, to support the discussion.

3.

Results

Fish were present in ail sampling sites (n = 554) while EPTC were recorded in 147, 137, 149 and 138 sampling sites out of 155. 75% of fish species and 55% of insect species were cornmon, commonness being 59%, 84%,41% and 44% within the insect groups E, P, T and C, respectively.

The fish madel explained 58.1% of the total variance in numbers of common species, as estimated by the deviance of the final madel (403.37) and that of the null madel (961.70). Slope was negatively correlated with numbers of common species, and distance from the source was positively correlated with fish commonness (p < 0.0001; Table 1). Other factors under consideration (elevation, stream arder) did not correlate significantly with the number of common fish species. In E, P and C, commonness increased with elevation (p = 0.001, p < 0.0001 and p = 0.05, respectively) but not with ether factors. The corresponding models explained 77.4%, 71.6% and 61.9% of the total variance in numbers of common species, respectively. In T, commonness increased with beth elevation and distance from the source (p < 0.0001 and p = 0.002, respectively). The madel explained 80.0% of the total variance in numbers of common species, as estimated by the deviance of the final mo del (47 .90) and that of the null madel (239.45). In sum, the GLM showed that commonness in fish and aquatic insects can be related to the downstream location of sampling sites within a large stream system, and that biodiversity patterns of stream animais may emerge from the distributions of beth common and rare species or mostly from the distribution of common species, depending upon the considered taxa.

The correlation coefficients between each of the subassemblage richness patterns and the richness pattern produced by ali species increased in different ways along the Backwards models with Log 10 -transformed data. Only variables with p < 0.05 are interpreted as statistically significant. For variables not included in the models no parameter estimate is presented and the F and p values correspond to the values when added to the final models.

Deviances of the null models, dispersion values (q> =deviance of the final modeVdegrees of freedom) and percentage of variance explained are given.

common-to-rare and rare-to-common sequences, whatever the taxa under consideration {Fig. 2a-e). However, for a given number of species, correlations were always greater for the

セ @ ü 0.2 セ セ M N L N N N L ⦅ N j N ケ ᄋ @ P セ P @ 0.2 0.4 0.6 0.8 1.0 1.0 . -"' 0.8 .. JI c 0 . M セ @ 0.6 . " Qi '!1." t: 0.4 0 セ O @ ü 0.2 M B セ @ ᄋ セ @ ッ セ N ッ @
0.2 0.4 0.6 0.8 1.0 Proportion of assemblage former than for the latter. Thus, common species gave a doser approximation to overall patterns of species richness than did rare species. For both T and C data sets, common species still produced a quicker rise in correlation than rare species with the increasing accumulated information, rather than numbers of species (Fig. 2i-j). For E and P, common and rare species produced a similar rise in correlation with the increasing accumulated information (Fig. 2g-h). Last, for the fish data set, the rarer species became more strongly correlated with increasing accumulated information (Fig. 2f).

Discussion

The biodiversity patterns of stream animais may emerge from convergence in the distributions of both common and rare species (fish), or mostly from the distribution of common species (aquatic insects), as swnmarized in Fig. 3. We describe biodiversity patterns through upto downstream gradients, with common species approximating well the overall pattems of species richness. It is worth noting that the variance explained by our Gl.Ms was about 58% in common fish and 62-80% in common insects, although we used a limited number of physical variables. The suggested schemes are thus likely to perform wellin ether areas. In fish, both the numbers of common (this study) and rare species [START_REF] Céréghino | Using self-organizing maps ta investigate spatial patterns of nonnative species[END_REF] increase towards downstream areas, as a result of downa stream additions of species. In aquatic insects (Fig. 3b), the overall EPTC richness was reported to peak in the intermediate section of the downstream continuum of the study area {Céréghino et al, 2003). However, rarity increased with distance from the source, or declined with elevation [START_REF] Céréghino | Using self-organizing maps ta investigate spatial patterns of nonnative species[END_REF]. The analyses carried out in the present work suggest that the number of common Trichoptera species peaks in the intermediate section of the downstream continuum, because beth elevation and distance from the soUit:e {which are negatively correlated) were correlated to commonness. In Ephemeroptera and Coleoptera, commonness increased with elevation but not with ether factors. Commonness also increased with increasing elevation in Plecoptera, although this group is rather confined to the upper mountainous sections of the stream system. Therefore, common Trichoptera would have the greatest influence on the patterns of commonness in EPTC, and, subsequently, on the overall EPTC richness pattems. Whilst there are a number of studies showing that there are generally a few taxa making up most of the community composition (e.g., [START_REF] Nijboer | Rare and common macroinvertebrates: definition of distribution classes and their boundaries[END_REF], the number of common species in our study area may seem quite high, particularly in aquatic insects. This observation could have two explanations, which are not mutually exclusive. First, the total number of sampling sites was lower for EPTC {155) than for fish {554), which means that an insect species had to be found at eight sites to be "'common•, whereas for tish a species had to be found at 28 sites out of 554 to be assigned as common. Second, des pite a possible bias in our perception of common EPTC species, our sampling area represents three biogeographical regions of the Limnofauna Europea [START_REF] Illies | Limnofauna Europaea[END_REF]: Pyrenees Mountains in the southem part of the stream system, Massif Central Mountains in the eastern part, and occidental plains in northwestem areas. A preliminary analysis of EPTC species distributions showed that species compositions allowed for a clear distinction of these regions in our study area [START_REF] Céréghino | Spatial analysis of stream invertebrates distribution in the Adour-Garonne drainage basin {France), using Kohonen self organizing maps[END_REF]. These marked biogeographical differences could thus account {at least partially) for the high number of common insect species. Random inc:lusion of species is directly linked to the sampling effort. The greater the sample size, the more individuals and the more rare species one should find at a given site. Nevertheless, our sampling effort was consistent (both for fish and EPTC), while the number of captured individuals followed similar trends as the species richness patterns in relation to the upstream-downstream gradient of environmental conditions. These settings thus allowed us to compare the sites to understand the relationships between the location of sites within the stream system, and commonness or rarity .

Cl) Q) M セ @ o. Cl) 0 03 .c E ::J z b Cl) Q) ë3 Q) o.
It should be noted that aquatic insects were collected with a surber sampler. Surber samplers are widely used for freshwater monitoring. but they can fail to capture the non-benthic invertebrates. However, this limitation is particularly relevant for downstream sections, i.e. where lower current velocities and deeper water allow the development of macrophytes and the presence of swimmers. For instance, Coleoptera are likely to occur in microhabitats such as the vegetation or the water column. In upstream sections (mountain areas, fast flowing streams on rocky substrates), almost ali EPTC are benthic. Although we were really working only with the benthic fraction of these taxa diversity, surber samplers remain rele-vant in quantitative analyses of stream organisms because they allow to standardize data collection, and thus to compare taxa diversity in space and time [START_REF] Resh | Rapid assessment approaches to biomonitoring using benthic macroinvertebrates[END_REF].

A challenge of recent research is to assess models having the broadest capability of predicting patterns of river community organization (e.g., the European Water Framework Directive; Sachon and Wassen, 2002). The River Continuum Concept {RCC; [START_REF] Vannote | The river continuum concept[END_REF]) offered a "reference scheme" describing the continuous gradient of physical characteristics of streams from headwaters to mouth and the resulting functional responses under natural conditions. However, there is little chance to find a river which fits the RCC along its whole course {Campin and [START_REF] Campin | Spatial patterns of macroinvertebrate functional feeding groups in streams in relation ta physical variables and land-caver in Southwestern France[END_REF], and this situation raises concerns about the possibility to identify broadly applicable patterns (e.g., as those suggested in Fig. 3), based on relevant structural attributes of aquatic communities on a large spatial scale. Specifically, rivers are increasingly affected by anthropogenic disturbance, resulting in modifications of their physical and chemical conditions, disruption of natural dispersal pathways {Kruk and [START_REF] Kruk | Impoundment impact on populations of facultative riverine fish[END_REF], and, subsequently, in changes within animal communities. In most cases, alterations of river habitats lead to lasses of taxa, and spatial discontinuities in predictable downstream gradients [START_REF] Campin | Sensitivity of aquatic insect species richness ta disturbance in the Adour-Garonne stream system {France)[END_REF]. Explicit schemes, such as distribution patterns of rare and common species within large watersheds, are therefore of obvious value to support field research and/or hypothesis testing, as well as integrated management directives.

The grea ter contribution of common species to patterns of species richness in all taxonomie groups calls for mechanistic explanations, to better understand why common species concentrate in sorne areas and occur in law numbers in ethers. Indeed, if significant correlations were generated between fish or insect species richness (rare, common), and distance from the source and/or elevation, the latter does not reflect direct causal relationships. The literature supports the idea that temperature plays an important role in explaining the relationship between elevation or distance from the source (i.e., the downstream gradient) and fish assemblages (e.g., [START_REF] Bardonnet | Influence of daily variations of light and temperature on the emergence rhythm of grayling fry (Thymallus thymallus)[END_REF]. Temperature governs the species population dynamics through growth and fecundity [START_REF] Lobon-Cervia | The reproductive tactic of dace in central Siberia: evidence for temperature regulation of the spa tic-temporal variability of its life history[END_REF] , by acting as a physicochemical habitat filter {Poff, 1997) with respect to species traits such as metabolism, energetic demands and reproduction. Other underlying factors related to the downstream gradient may explain site to site differences in fish species richness. Dawnstream addition of fish species (rare, common, overall) may result from increased living space in larger stream sections, increased habitat diversity (e.g., access to floodplain habitats and backwaters), and grea ter habitat stability such as reduced flow variability {Schlosser, 1987; [START_REF] Rahel | Fish assemblages and habitat gradients in a Rocky Mountain-Great Plains stream: biotic zonation and additive patterns of community change[END_REF]. Conversely, headwater streams with uniform habitat of law volume and shallow pools faveur small fishes [START_REF] Schlosser | A conceptual framework for fish communities in small warmwater streams[END_REF]. From lithophilous to phytophilous reproduction guilds {Balan, 1975), fish require specifie substrates for egg deposition, the quality and heterogeneity of which is strongly dependent on the downstream location of sites {Penczak and [START_REF] Penczak | The impact of stream order on fish populations in the Pilica drainage basin, Poland[END_REF][START_REF] Reyjol | Longitudinal diversity patterns in streams: comparing invertebrates and fish communities[END_REF]. Furthermore, Oberdorff et al. {2002) reported a decrease in invertivorous species richness and abun-dance and an increase in omnivorous species richness and abundance from upstream to downstream areas. Severa! abiotic factors may influence benthic diversity (e.g., flow regime, water temperature and chemistry, substrate nature and composition, and food resources), and the relative importance of these factors is always difficult to assess because they interact in nature. However, the diversity of aquatic insect communities strongly depends on the diversity and stability of stream habitats {Cummins, 1979; [START_REF] Vinson | Biodiversity of stream insects: variation at local, basin, and regional scales[END_REF][START_REF] Heino | Defining macroinvertebrate assemblage types of headwater streams: implications for bioassessment and conservation[END_REF], which provide the ecological niches for the development of species {Malmqvist and Otto, 1987). Specifically, stream flow determines the spatiotemporal variability of physical microhabitats (Moog and ]anecek, 1991;[START_REF] Robinson | Influence of flow regime on life history, production, and genetic structure of Baetis tricaudatus (Ephemeroptera) and Hesperoperla paciftca (Piecoptera)[END_REF], thus influencing the benthic community structure through the prominent role of substrate size and composition, current velocities, and bed scour. Therefore, the higher flow variability observed in intermediate (mountainous-piedmont) sections of the river continuum certainly has a positive influence on habitat heterogeneity, to the benefit of most insect species (i.e., those common species, which create most of the spatial structure in richness patterns of aquatic insects).

Many species in biological communities are rare, and this is true of aquatic animal communities {Marchant et al., 1999). Common species made a grea ter contribution to species richness in all data sets, and this is often true of biological communities {Pearman and Weber, 2007). However, in sorne taxa, the rarer species became almost equally {Ephemeroptera, Plecoptera), or more strongly correlated (fish) with species richness when increasing information along the common-to-rare and rare-to-common sequences. These patterns suggest that these rarer insect or fish species show a similar or stronger affinity, on a species-for-species basis, for high richness areas than do the commoner species. Such an assumption also fits with the observation that areas which carry more rare species (e.g., large and high-ordered streams in our study) may also concentra te an important fraction of the regional biodiversity [START_REF] Céréghino | Using self-organizing maps ta investigate spatial patterns of nonnative species[END_REF]. Finally, throughout the world, governmental action plans aim at delineating natural zones of ecological, faunistic and floristic interest (e.g., the Natura 2000 network in Europe). The first objective of such plans is to identify areas which concentrate conservation values, i.e., hot-spots of species richness, areas containing rare species or endangered habitats. Among the questions usually asked to the scientific experts, two are of particular interest: (i) which environmental variables may explain variations in numbers of species? and (ii) within a given regional system, which areas contain rare species? To address these concerns, distribution patterns of species richness, and rarity and commonness must in general be derived from environmental conditions. From our analyses, it appeared that: (i) simple variables such as distance from the source and elevation were able to predict variations in the numbers of species {bath rare and common) for fish and insects, respectively, and (ii) there was a marked concentration of rare fish and insect species in large and high-ordered streams. Finally, if common species approximated well the overall patterns in species richness, fish and aquatic insects showed different responses to the downstream gradients in physical variables, almost certainly because of differences in ecological requirements. Although our study illustra tes the importance of including bath vertebrate and invertebrate taxa in analysing animal biodiversity patterns, it remains one of a very few attempts to do so.

Fig. 1 -

 1 Fig. 1 -The Garonne stream system, and location of the sampling sites for (a) 6sh and (b) aquatic insecbl (Ephmmuptenl, Plecoptera, Trichoptera and Coleoptera).

Fig. 2 -

 2 Fig. 2 -Sequential correlation between partial and full assemblage spedea rtchness patterns. Each partial assemblage was grown by adding one spec:ies at a dme, correladng the richneaa pattern at each stage with that of the full assemblage of species. Species were added in rank orcier from least to most widespread (rare-m-common, black squares) and conversely (common-to-rare, grey squares). ln (a}-{e) correlations are plotted against the proportion of the total nwnber of species that are in the partial assemblage. In (f)-(j) correlations ue plotted against accumulated information in species distn"budons, a meaaure of the overall numbera of recorda of presence or absence (see Methods for details). Plots (a) and (f) are for fi.sh, (b) and (1) for Ephemeroptma, (c) and (h) for Plecoptma, (d.) and (i) Tric:hoptml, and (e) and (j) for Coleoptmr.

  Fig. 3 -Trends in the upstream-downstream patterns of rare, conunon and overall specles rlchness ln (a) fi.sh and (b) aquatlc lnaectll (Ephemeropoera, Plecoptena, nichopoera and O»leoptera combined) in large stream systems. The tnmda in nwnbers of raJe and common species were drawn from GLMs fatted with a log link func:tion (Santoul et al., 2005 and this study, respectively). Overall species richness patterns were interpreted from the distnbutions of :rare and common apeciea.

Table 1 -Models analysing the patterns of common species of the fi.ve freshwater taxa (Fish, Ephemeroptera, Plecoptera, Trichoptera and Coleoptera) distributions

 1

	Freshwater taxa	Effect	Estima te	F	df	p
	Fish	Stream order		0.09	1,414	0.77
		Elevation		0.01	1,412	0.92
		Slope	-0.4544	62.57	1,415	<0.0001
		Distance source	0.3811	39.42	1,415	<0.0001
		Deviance	961.70
		q,	0.73
		Variance explained	58.1%
	Ephemeroptera	Stream arder		0.00	1,87	0.97
		Elevation	0.6664	11.60	1,88	0.001
		Slope		0.06	1,87	0.81
		Distance source		1.58	1,87	0.21
		Deviance	283.82
		q,	0.51
		Variance explained	77.4%
	Plecoptera	Stream arder		0.18	1,87	0.67
		Elevation	1.4879	25.49	1,88	<0.0001
		Slope		1.29	1,87	0.26
		Distance source		3.18	1,87	0.08
		Deviance	641.22
		q,	1.38
		Variance explained	71.6%
	Trichoptera	Stream arder		1.19	1,86	0.28
		Elevation	1.2994	21.55	1,87	<0.0001
		Slope		0.00	1,86	0.99
		Distance source	0.5011	9.85	1,87	0.002
		Deviance	239.45
		q,	0.3770
		Variance explained	80.0%
	Coleoptera	Stream arder		0.29	1,87	0.59
		Elevation	0.4262	4.03	1,88	0.05
		Slope		0.00	1,87	0.99
		Distance source		3.27	1,87	0.07
		Deviance	304.45
		q,	0.76
		Variance explained	61.9%

Aclmowledgements

We wish to thank two anonymous reviewers who provided valuable comments on an earlier version of this paper.


Eric Huaqiang Shu

Pierre Simon

Laurent Ros

E P Simon

Third-order Kalman Filter : tuning and steady-state performance

Keywords:

This letter deals with the Kalman filter (KF) based on a third-order integrated random walk model (RW3). The resulting filter, noted as RW3-KF, is well suited to track slow time-varying parameters with strong trend behaviour. We first prove that the RW3-KF in steady-state admits an equivalent structure to the third-order digital phase-locked loops (DPLL). The approximate asymptotic mean-squared-error (MSE) is obtained by solving the Riccati equations, which is given in a closedform expression as a function of the RW3 model parameter: the state noise variance. Then, the closed-form expression of the optimum state noise variance is derived to minimize the asymptotic MSE. Simulation results are given for the particular case where the parameter to be estimated is a Rayleigh channel coefficient with Jakes' Doppler spectrum.

I. INTRODUCTION

Kalman filters (KF) are commonly used to track timevarying parameters. The applications of KF cover a various range of systems, like GPS systems [START_REF] Kaplan | Understanding GPS: Principles and Applications, Second Edition[END_REF], Multi-carrier systems [START_REF] Hijazi | Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains[END_REF], MIMO systems [START_REF] Komninakis | Multi-input multi-output fading channel tracking and equalization using Kalman estimation[END_REF], etc. The design of KF requires a linear recursive state-space representation of the parameter to be observed. The most used approximation model, especially for channel estimation problems, is the first-order Auto-Regressive model (AR1), combined with either a correlation matching (CM) criterion for the fast time-varying scenario [START_REF] Hijazi | Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains[END_REF] [START_REF] Simon | Joint Carrier Frequency Offset and Channel Estimation for OFDM Systems via the EM Algorithm in the Presence of Very High Mobility[END_REF], or a minimum asymptotic variance (MAV) criterion for the slow varying scenario [START_REF] Ghandour-Haidar | On the use of first-order autoregressive modeling for Rayleigh flat fading channel estimation with Kalman filter[END_REF], to fix the AR coefficient. However, in certain systems, the parameter to be estimated exhibits strong trend behaviour, and the use of second-order or higher-order models is more suitable than a first-order model. For example in a satellite receiver, third-order KF as well as third-order DPLLs are often used to tackle the problem of phase tracking in the presence of time-varying Doppler frequency offset [START_REF] Kaplan | Understanding GPS: Principles and Applications, Second Edition[END_REF]. However, the tuning and performance of these estimators are most often obtained from simulation or empirical results.

In this paper, we provide analytic results about the optimal tuning and the steady-state performance of a KF based on a RW3 model. For that, we first prove that this third-order KF has the same structure in a steady-state mode as a specific equal-order DPLL, hence extending the results of [6] [7] obtained for a second-order KF.

Section II gives the approximation model and the formulae of RW3-KF. In section III, we analyze and optimize the asymptotic MSE of RW3-KF. Section IV validates the analysis and assumptions by means of MSE and BER (bit error rate) simulations, the first-order AR model-based KFs (combined with CM and MAV criterion, respectively noted as AR1 CM -KF and AR1 M AV -KF) are selected as references.

II. STATE-SPACE MODEL AND KALMAN FILTER

Assume the parameter to be estimated α is a zero-mean circular complex process with variance σ 2 α . The variable α is supposed to be a narrow-band stationary process, with a Power Spectrum Density (PSD) Γ α (f ) with a support limited within ±f d . We consider the RW3 model as an approximation of the time-variation of α:

α(n) = α(n-1) + δ (n-1) + 1 2 ξ (n-1) , (1)
δ (n) = δ (n-1) + ξ (n-1) , (2)
ξ (n) = ξ (n-1) + u (n) , (3)
where u (n) is the state noise, a zero mean complex state noise with variance σ 2 u . The model is updated at sample rate. The time interval between each sample, T , represents a unit delay. A simplistic observation model is used 1 :

y (n) = α (n) + w (n) , (4)
where w (n) is a zero-mean additive white noise with variance σ 2 w . The dynamic evolution equations (1)-(3) and the observation equation ( 4) compose the state-space model of α (n) . The on-line unbiased estimation α(n) can be carried out by KF. The MSE σ 2

ǫ def = E ǫ (n) 2 of the estimation error ǫ (n) def = α (n) -α(n) will be investigated.
Rewrite the state-space model in the matrix form:

a (n) = Ma (n-1) + u (n) , (5)
y (n) = Sa (n) + w (n) , (6)
with the state vector a

(n) = α(n) δ (n) ξ (n)
T , the state noise vector u (n) = 0 0 u (n) T , the selection vector S = 1 0 0 and the evolution matrix

M =   1 1 1 2 0 1 1 0 0 1   , the
RW3-KF could then described by two-stage equations:

Time Update Equations â(n|n-1) = Mâ (n-1|n-1) , ( 7)

P (n|n-1) = MP (n-1|n-1) M T + U, (8)
Measurement Update Equations

K (n) = P (n|n-1) S T SP (n|n-1) S T + σ 2 w , (9)
â(n|n) = â(n|n-1) + K (n) (y (n) -Sâ (n|n-1) ), (10)
P (n|n) = (I -K (n) S)P (n|n-1) , (11)
with the Kalman gain

K (n) = k 1(n) k 2(n) k 3(n) T , the state noise variance matrix U =   0 0 0 0 0 0 0 0 σ 2 u   , P (n|n-1)
and P (n|n) are respectively the covariance matrices of the prediction error and the estimation error.

III. ASYMPTOTIC MSE ANALYSIS

A. Steady-state RW3-KF

Since the linear model (( 5),( 6)) is observable and controllable, an asymptotic regime is quickly reached ( [8] Ch. 13.3). In other words, P (n|n) ,P (n|n-1) and K (n) converge to constant values when n is large enough, i.e., Note that P (∞) and P ′ (∞) are real symmetric matrices. This can be easily verified from ( 8), ( 9), [START_REF] Winkelstein | Closed form evaluation of symmetric two-sided complex integrals[END_REF] if the KF starts with a real-valued matrix P (0|-1) . K (∞) is also a real vector. In the steady state, from ( 7) and [START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF], the recursive equations of RW3-KF can be reduced to a time-invariant filter:

K (n) = K (n+1) = K (∞) def = k 1 k 2 k 3 T , (12)
P (n|n) = P (n+1|n+1) = P (∞) def =
P (n|n-1) = P (n+1|n) = P ′ (
α(n|n) = α(n-1|n-1) + δ(n-1|n-1) + 1 2 ξ(n-1|n-1) + k 1 v ǫ(n) , (15) δ
(n|n) = δ(n-1|n-1) + ξ(n-1|n-1) + k 2 v ǫ(n) , (16) ξ(n|n) = ξ(n-1|n-1) + k 3 v ǫ(n) , ( 17
) with v ǫ(n) = y (n) -(α (n-1|n-1) + δ(n-1|n-1) + 1 2 ξ(n-1|n-1) ). (18)
Transforming ( 15), ( 16), (17) to Z-domain and substituting δ and ξ yield:

α(z)(1-z -1 ) = k 1 + (k 2 + 1 2 k 3 )z -1 1 -z -1 + k 3 z -2 (1 -z -1 ) 2 v ǫ (z).
(19) Combining ( 18), ( 15) and ( 4), and after Z-transform we have:

v ǫ (z) = 1 1 -k 1 • (α(z) -α(z) + w(z)), (20)
then substitute (20) into (19), we obtain the input-output equation:

α(z) = L(z) • α(z) + L(z) • w(z), (21)
with L(z) the transfer function of steady-state RW3-KF given in (22).

In the slow fading scenario (f d T ≪ 1), we are interested in the low frequency domain part of L(z) (f T ≪ 1), using the approximation 1 -z -1 ≈ pT , with z = e pT and p = j2πf . With such an approximation, the steady state transfer function of the RW3-KF L(e j2πf ) is equivalent to the typical transfer function of the third-order analog PLL ( [START_REF] Shan | Design and Implementation of Bandwidth Adaptable Third-Order All Digital Phase-Locked Loops[END_REF], eqn. ( 2), ( 4) and [START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF] eqn. ( 22)):

L(e pT ) ≈ (m + 2)ζωn • p 2 + (1 + 2mζ 2 )ω 2 n • p + mζω 3 n p 3 + (m + 2)ζωn • p 2 + (1 + 2mζ 2 )ω 2 n • p + mζω 3 n (23) with k1 = (m + 2)ζωnT + (1 + 2mζ 2 )(ωnT ) 2 + mζ(ωnT ) 3 1 + (m + 2)ζωnT + (1 + 2mζ 2 )(ωnT ) 2 + mζ(ωnT ) 3 , ( 24
) k2 = (1 + 2mζ 2 )(ωnT ) 2 + 3 2 mζ(ωnT ) 3 1 + (m + 2)ζωnT + (1 + 2mζ 2 )(ωnT ) 2 + mζ(ωnT ) 3 , ( 25
) k3 = mζ(ωnT ) 3 1 + (m + 2)ζωnT + (1 + 2mζ 2 )(ωnT ) 2 + mζ(ωnT ) 3 , (26)
and with m the capacitance ratio, ζ the damping factor, ω n = 2πf n the natural radian frequency of the loop. They are real positive physical parameters, and ω n T ≪ 1 when assuming a slow reaction of the filter. A useful inequality could be obtained with

0 < k 3 ≪ k 2 ≪ k 1 < 1. ( 27
)
This inequality is obtained by comparing the numerators of (24), (25), (26), using ω n T ≪ 1.

We aim to find the relation between the Kalman gains k 1 , k 2 , k 3 , and the state noise variance σ 2 u , in order to optimize the estimation error σ 2 ǫ with respect to σ 2 u . From (9), 28), (29) compose the so called Riccati equations.

L(z) = (k 1 -k 2 + k3 2 )(1 -z -1 ) 2 + (k 2 -k3 2 )(1 -z -1 ) + k 3 (1 -k 1 )(1 -z -1 ) 3 + (k 1 -k 2 + k3 2 )(1 -z -1 ) 2 + (k 2 -k3 2 )(1 -z -1 ) + k 3 . ( 22
)
By solving these equations, we could find the expressions of the elements of P ′ (∞) as a function of k 1 , k 2 , k 3 , and σ 2 u . To this end, after some manipulations on (29), we first find:

P ′ 13 = P ′ 31 = σ 2 u k 3 , (30)
which enables us to find after some manipulations:

P ′ 11 = 8k 2 + k 3 (6k 1 + 3k 2 + k 3 ) 2k 2 3 (2k 1 + 2k 2 + k 3 ) σ 2 u . (31)
Then, a relation between k 1 , k 2 and k 3 is found:

k 2 2 = 2k 1 k 3 . ( 32
)
In the sequel, it is assumed that P ′ 11 ≪ σ 2 w , which means that the Kalman gain is low k 1 ≪ 1, according to (28). Then we deduce k 1 from (28), k 3 from (28) and (30) and k 2 from (32) respectively, that is:

k 1 ≈ P ′ 11 σ 2 w , k 3 = σ u P ′ 11 + σ 2 w ≈ σ u σ w , k 2 ≈ 2P ′ 11 σ u σ 3 w . ( 33
)
To further simplify the calculation, we apply the approximation ( 27) on (31), yielding:

P ′ 11 ≈ 2k 2 k 1 k 2 3 σ 2 u . ( 34
)
By combining (33), (34), P ′ 11 can be expressed as a function of σ u and σ w :

P ′ 11 ≈ 2σ 1 3 u σ 5 3 w = 2σ 2 w ( σ u σ w ) 1 3 , (35)
and finally,

k 1 ≈ 2( σ u σ w ) 1 3 , k 2 ≈ 2( σ u σ w ) 2 3 = k 2 1 2 , k 3 ≈ σ u σ w = k 3 1 8 . ( 36
)
Using the approximated Kalman gain relation 0 < k 3 ≪ k 2 ≪ k 1 ≪ 1, the transfer function of RW3-KF (22) can be simplified as:

L(z) ≈ k 1 (1 -z -1 ) 2 + k 2 (1 -z -1 ) + k 3 (1 -z -1 ) 3 + k 1 (1 -z -1 ) 2 + k 2 (1 -z -1 ) + k 3 .
(37) Comparing (37) and (23), we get 3 . Then by using (36), we obtain m = 2, ζ = 0.5, while its natural radian frequency ω n T can be tuned as k1 2 , or eventually ( σu σw )

k 1 ≈ (m + 2)ζω n T , k 2 ≈ (1 + 2mζ 2 )(ω n T ) 2 , k 3 ≈ mζ(ω n T )
1 3 . Thus, we can conclude that the RW3-KF is equivalent in steady-state mode and slow-tracking scenario to the third-order DPLL with fixed given parameters (m = 2, ζ = 0.5). This conclusion generalizes to the third-order the connection between DPLL and KF established in [START_REF] Driessen | DPLL bit synchronizer with rapid acquisition using adaptive Kalman filtering techniques[END_REF] [START_REF] Patapoutian | On phase-locked loops and Kalman filters[END_REF] for the second-order.

B. Mean Squared Error Analysis

The (unbiased) estimation error is defined by:

ǫ(z) = α(z) -α(z) = (1 -L(z)) • α(z) -L(z) • w(z) (38)
and the mean squared error is thus composed by two parts:

σ 2 ǫ = E{ǫ • ǫ * } = σ 2 ǫα + σ 2 ǫw , (39)
σ 2 ǫw is the static error variance which results from the channel noise w, whereas σ 2 ǫα is the dynamic error variance, which results from the parameter α variations.

The static error variance is developed as:

σ 2 ǫw = σ 2 w • T + 1 2T -1 2T |L(e j2πf T )| 2 df B L = 5 2 k 1 -1 4 k 2 1 -5 8 k 3 1 -5 64 k 4 1 3 -9 8 k 2 1 -9 32 k 3 1 -3 64 k 4 1 σ 2 w ≈ 5 3 k 1 3 3 σ 2 w , (40)
where the integral term B L is the equivalent noise bandwidth. It can be calculated by the method presented in [START_REF] Winkelstein | Closed form evaluation of symmetric two-sided complex integrals[END_REF]. Note that we have applied the condition 0 < k 1 ≪ 1 for the approximation. The dynamic error variance is developed as:

σ 2 ǫα = + 1 2T -1 2T Γ α (f ) • |1 -L(e j2πf T )| 2 df ≈ + 1 2T -1 2T Γ α (f ) • (2πf T ) 6 k 2 3 df = (2π) 6 k 2 3 S α , (41)
where 6 df is the term which contains the PSD of α. For the reason of simplicity, we apply e -j2πf T ≈ 1 -j2πf T as well as

S α = + 1 2T -1 2T Γ α (f )(f T )
2πf d T ≪ k 1 3 3 ≈ ω n T ≪ 1 to calculate |1 -L(e j2πf T )| 2 in
the slow variation channel case. The global MSE is then obtained by combining (40) and (41). After substituting the approximation (36) for k 3 , the objective function to optimize is given by:

σ 2 ǫ = 5 3 σ 5 3
w σ

1 3 u + (2π) 6 S α σ 2 w σ 2 u . (42)
The minimization can be done by imposing the partial derivative of global MSE σ 2 ǫ equal to 0, yielding:

σ 2 u opt = (2π) 36 • ( 18 5 S α ) 6 • σ 2 w 1 7 , (43)
and the corresponding minimized MSE is:

σ 2 ǫ min = 7 • 5 9 π • σ 2 w 6 7
• S

1 7 α . (44)
C. An application to the estimation of Rayleigh channel with Jakes' Doppler spectrum From (43) and (44), we note that the optimum parameter and the corresponding minimized MSE could be computed whatever the channel PSD is. Now we take the estimation of Rayleigh channel with Jakes' Doppler spectrum as an example. The PSD of α is defined as:

Γ α (f ) =    σ 2 α πf d 1-f f d 2 , for |f | < f d , 0, for |f | ≥ f d . (45)
A variable change cosθ = f f d is applied to calculate the integral S α and we have:

S α = +f d -f d (f T ) 6 • σ 2 α πf d 1 -f f d 2 df = 5 16 (f d T ) 6 • σ 2 α . ( 46
)
The optimal σ 2 u and the corresponding minimized MSE are then obtained directly:

σ 2 u opt = 3 12 2 18 • (σ 2 α ) 6 • σ 2 w • (2πf d T ) 36 1 7 , (47)
σ 2 ǫ min = 35 16 • ( 16 9 πf d T • σ 2 w ) 6 7 • (σ 2 α ) 1 7 . (48)

IV. SIMULATION RESULTS AND CONCLUSION

The MSE analysis is verified by Monte-Carlo simulations over a Rayleigh flat fading channel. Fig. 1 shows the MSE of AR1 CM -KF [START_REF] Hijazi | Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains[END_REF] [3] [START_REF] Simon | Joint Carrier Frequency Offset and Channel Estimation for OFDM Systems via the EM Algorithm in the Presence of Very High Mobility[END_REF], AR1 M AV -KF [START_REF] Ghandour-Haidar | On the use of first-order autoregressive modeling for Rayleigh flat fading channel estimation with Kalman filter[END_REF] and RW3-KF as a function of SNR, with f d T = 10 -3 . The theoretical MSE of RW3-KF as well as the online BCRB (Bayesian Cramer-Rao Bound) [START_REF] Hijazi | Bayesian Cramer-Rao bounds for complex gain parameters estimation of slowly varying Rayleigh channel in OFDM systems[END_REF] are used as references. Fig. 2 shows the MSE of these estimators as a function of f d T with fixed SNR=20dB. From Fig. 1 and2, we find that the theoretical and the simulation lines of RW3-KF approximately coincide. The MSE of RW3-KF is proportional to the 6 7 power of noise variance σ 2 w (thus inversely proportional to the SNR), and is also proportional to the 6 7 power of f d T . On the other hand, compared to the AR1 CM -KF, the AR1 M AV -KF has a much improved asymptotic performance, which means that the MAV criterion is a better choice for computing the AR1 coefficient. However it is still far from the lower bound due to the loworder filtering that causes the loss of dynamic information. Meanwhile, the RW3 model fits the real channel much better than the AR models in the slow fading case. Moreover, the MSE of RW3-KF is very close to the online BCRB.

For the BER simulation, we use QPSK transmitted symbols. The estimation is in semi-blind mode, that is, the data block is composed of 20 pilot symbols followed by 180 unknown symbols (for which the KF is in decision-directed mode). Fig. 3 shows the simulation result, where we can observe that with the optimized σ 2 u opt , the RW3-KF attains a performance close to the one with perfect channel knowledge.

To conclude, we have discussed in this letter the thirdorder modeling of the Kalman Filter for parameter estimation problems, where an application to Rayleigh fading channel with Jakes' spectrum was also introduced. The explicit formulae of the optimum parameter and the asymptotic MSE of the RW3-KF were given, assuming the knowledge of the channel statistics. A connection between the steady-state RW3-KF and the typical third-order DPLL was established. We also conclude that, for KF-based estimators, the well-tuned third-order random walk model is more adequate compared with the first-order AR model in the low-variation context, with the resulting estimator performance very close to the BCRB. Possible future directions are to extend this work to the vectorial case for multi-path channel and/or multi-carrier modulation scenarios. Also, MSE performance of the other components of the RW3 model could be investigated.

From ( 11 P 11 P 11 ( 1 -k 1 )P ′ 12 ( 1 -k 1 11 =

 1111111121111 ),[START_REF] Kay | Fundamentals of Statistical Signal Processing : Estimation Theory[END_REF] and by using the symmetry of P (∞) and P ′ 12 P 13 P 12 P 22 P 23 P 13 P 23 P 33 P 11 +2P

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

 123 Fig. 1: MSE versus SNR with f d T = 10 -3

  12 +P 13 +P 22 +P 23 + 1 4 P 33 ; P

	′ 12 = P 12 + 22 = P 22 + 2P 23 + P 33 ; P 2 P 33 ; P 2 P 23 + P 13 + 1 P 22 + 3 ′ ′ 13 = P 13 + P 23 + 1 2 P 33 ; P

′ 23 = P 23 + P 33 ; P ′ 33 = P 33 + σ 2 u . The equations (

This model is adequate for many applications, e.g. it could be a flat fading channel model, α is then the complex amplitude of channel; or in the vehicle tracking problem, α could be in matrix form, composed by the position coordinates and velocities of vehicle. etc.