J Rydberg
email: johan.rydberg@emg.umu.se

P Rosén

L Lambertsson

F De Vleeschouwer

S Tomasdotter

R Bindler

Assessment of the spatial distributions of total-and methyl-mercury and their relationship to sediment geochemistry from a whole-lake perspective

The aim of this study was to determine the spatial variability for total-and methylmercury in surface sediments (0-2 cm) across a single whole-lake basin, and to relate this variability to the sediment's geochemical composition. 83 surface sediment samples from Stor-Strömsjöna lake with multiple sub-basins located in northern Swedenwere analyzed for geochemical composition as well as total-mercury (total-Hg) and methylmercury (methyl-Hg; 35 samples) concentrations. Our results indicate that variations in fine-grained mineral matter (36%) and organic matter (34%) explain an equal amount of the total-Hg variation, but that their relative importance varies between different parts of the lake. Total-Hg concentrations were similar in locations controlled by organic matter or fine-grained mineral matter (average 109 ng g À1 ); however, total-Hg inventories (mass per unit area) were significantly higher in the latter (35 and 53 mg m À2 , respectively). Methyl-Hg concentrations are largely (55% of variance) controlled by water depth and sulfur concentration, which supports the importance of within lake methylation reported from other studies. Both for concentrations and inventories the spatial distribution for methyl-Hg in surface sediments is patchy, and interestingly the highest methyl-Hg inventory (1.4 mg m À2 ) was found in a shallow location with coarse-grained minerogenic sediment (very low organic matter). A large spatial variability, even within a single lake, is something that needs to be recognized, e.g., when studying processes affecting mercury cycling, mercury loadings and when using lake sediments to reconstruct historic mercury deposition.

Introduction

[2] Mercury (Hg) and its cycling has been studied from a wide range of different perspectives, and the research community currently possesses a broad knowledge on the behavior of Hg in the environment. In brief: Hg is largely added to the environment through atmospheric deposition upon the terrestrial landscape [START_REF] Hines | Mercury inputs and outputs at a small lake in northern Minnesota[END_REF][START_REF] Lindqvist | Mercury in the Swedish environment-Recent research on causes, consequences and corrective methods[END_REF]. Because Hg has a high affinity for organic matter, Hg storage in soils and transport to streams and lakes are largely coupled with that of organic matter [START_REF] Gabriel | Principal biogeochemical factors affecting the speciation and transport of mercury through the terrestrial environment[END_REF][START_REF] Skyllberg | Complexation of mercury(II) in soil organic matter: EXAFS evidence for linear two-coordination with reduced sulfur groups[END_REF]. Upon entering lakes a large part of the Hg is accumulated in sediments [START_REF] Hines | Mercury inputs and outputs at a small lake in northern Minnesota[END_REF]; these sediments can be used as archives to reconstruct Hg deposition history [Bindler et al., 2001a;[START_REF] Fitzgerald | The case for atmospheric mercury contamination in remote areas[END_REF][START_REF] Phillips | Increased mercury loadings to western Canadian alpine lakes over the past 150 years[END_REF][START_REF] Swain | Increasing rates of atmospheric mercury deposition in midcontinental North America[END_REF] and to calculate Hg budgets for the lake and its catchment [START_REF] Fitzgerald | Modern and historic atmospheric mercury fluxes in northern Alaska: Global sources and Arctic depletion[END_REF][START_REF] Yang | Mercury and lead budgets for Lochnagar, a Scottish mountain lake and its catchment[END_REF]. Under anoxic conditions, for example, in mires, hypolimnic waters and sediments, Hg can be methylated [START_REF] Branfireun | In situ sulphate stimulation of mercury methylation in a boreal peatland: Toward a link between acid rain and methylmercury contamination in remote environments[END_REF][START_REF] Eckley | Determination of mercury methylation potentials in the water column of lakes across Canada[END_REF][START_REF] Gilmour | Sulfate stimulation of mercury methylation in fresh-water sediments[END_REF]. The methylated Hg can then readily enter the food-web and be enriched in higher trophic levels [START_REF] Benoit | Geochemical and biological controls over methylmercury production and degradation in aquatic ecosystems[END_REF][START_REF] Ullrich | Mercury in the aquatic environment: A review of factors affecting methylation[END_REF]. However, even with this extensive knowledge it has proven difficult to fully account for variations in Hg and methyl-Hg (MeHg) within and between lakes, both in sediments and in fish [START_REF] Rypel | Mercury in southeastern U.S. riverine fish populations linked to water body type[END_REF][START_REF] Wiener | Mercury in soils, lakes, and fish in Voyageurs National Park (Minnesota): Importance of atmospheric deposition and ecosystem factors[END_REF].

[3] One aspect that is rarely fully considered is spatial variability in total-Hg and methyl-Hg accumulation within single lakes. Except for lakes with very small catchment to lake area ratios, Hg is largely added to the environment through deposition of inorganic Hg on the terrestrial part of a lake's catchment, and over time some of this Hg is transported from the catchment to the lake [START_REF] Hines | Mercury inputs and outputs at a small lake in northern Minnesota[END_REF][START_REF] Lindqvist | Mercury in the Swedish environment-Recent research on causes, consequences and corrective methods[END_REF]. In the boreal landscape this transport is generally assumed to be tightly coupled with the transport of organic matter [START_REF] Benoit | Geochemical and biological controls over methylmercury production and degradation in aquatic ecosystems[END_REF][START_REF] Gabriel | Principal biogeochemical factors affecting the speciation and transport of mercury through the terrestrial environment[END_REF][START_REF] Skyllberg | Complexation of mercury(II) in soil organic matter: EXAFS evidence for linear two-coordination with reduced sulfur groups[END_REF]. Upon entering the lake a governing assumption for the within-lake distribution of organic matter, and hence also Hg, is that it is controlled by sediment focusing [START_REF] Likens | Post-glacial history of Mirror Lake and its watershed in New Hampshire, U.S.A. An initial report[END_REF]. This generalized model for sedimentation describes a preferential transport of light, fine-grained material such as organic matter from shallower to deeper areas of a lake [START_REF] Håkanson | Principals of Lake Sedimentology[END_REF][START_REF] Likens | Post-glacial history of Mirror Lake and its watershed in New Hampshire, U.S.A. An initial report[END_REF], and the main drivers are considered to be basin slope, water depth and wave action [START_REF] Blais | The influence of lake morphometry on sediment focusing[END_REF][START_REF] Rowan | Estimating the mud deposition boundary depth in lakes from wave theory[END_REF]. However, not all lakes seem to fully comply with this generalized model. In part this can be explained by catchment characteristics having a significant effect on the quality and amount of the organic matter accumulated at nearshore locations [START_REF] Korsman | Spatial variability in surface sediment composition characterised by near-infrared (NIR) reflectance spectroscopy[END_REF]. It has also been shown that shallow-water vegetation [START_REF] Benoy | Sediment accumulation and Pb burdens in submerged macrophyte beds[END_REF], complex basin morphometry (e.g., multiple basins and islands) and prevailing wind directions [Bindler et al., 2001b] can have a strong influence on the spatial distribution of organic matter as well as Hg and lead (Pb).

[4] Because methylation is primarily a microbially mediated process, the formation of methyl-Hg in the sediment is determined by, for example, available Hg, number of viable methylating bacteria cells, presence of suitable substrate and the chemical conditions [START_REF] Benoit | Geochemical and biological controls over methylmercury production and degradation in aquatic ecosystems[END_REF][START_REF] Ullrich | Mercury in the aquatic environment: A review of factors affecting methylation[END_REF]. Hence, if the sediment -both its organic and inorganic fractions -is spatially variable then methylation, and thus, methyl-Hg concentrations will also likely vary greatly across the lake bottom. This spatially variable in situ net production of methyl-Hg will then be superimposed on the spatially variable inputs of methyl-Hg to the sediment from other sources (e.g., terrestrial or water column [START_REF] Branfireun | Speciation and transport of newly deposited mercury in a boreal forest wetland: A stable mercury isotope approach[END_REF][START_REF] Eckley | Determination of mercury methylation potentials in the water column of lakes across Canada[END_REF]). Taken together this suggests that the spatial distribution for methyl-Hg can be very complex in lakes, which might have important implications for Hg bioavailability in different parts of the lake [START_REF] Chételat | Habitat-specific bioaccumulation of methylmercury in invertebrates of small mid-latitude lakes in North America[END_REF].

[5] The spatial distribution of Hg in whole-lake basins has previously been studied both from the perspective of understanding total-Hg mass balance at a catchment or regional scale [START_REF] Fitzgerald | Modern and historic atmospheric mercury fluxes in northern Alaska: Global sources and Arctic depletion[END_REF][START_REF] Swain | Increasing rates of atmospheric mercury deposition in midcontinental North America[END_REF][START_REF] Yang | Mercury and lead budgets for Lochnagar, a Scottish mountain lake and its catchment[END_REF] and to investigate the relation between total-and methyl-Hg and other sediment variables, for example, organic matter, grain size, lake-basin morphometry, pH and redox conditions [START_REF] Croston | Spatial distribution and seasonal changes in methylmercury concentrations in shallow lakes[END_REF][START_REF] Ethier | Mercury empirical relationships in sediments from three Ontario lakes[END_REF][START_REF] Feyte | Mercury dynamics in lake sediments[END_REF][START_REF] Kainz | Mercury concentrations in lake sediments-Revisiting the predictive power of catchment morphometry and organic matter composition[END_REF][START_REF] Rhoton | Soil and sediment properties affecting the accumulation of mercury in a flood control reservoir[END_REF]. However, for most studies a relatively low number of sampling sites were used (n = 1-17), and most often these sites were distributed along a single transect from shallow to deep waters. The combination of a low sample number and the exclusion of, for example, shallow embayments introduce a risk of not capturing the true range of spatial variability in the studied lakes [cf. Bindler et al., 2001b;[START_REF] Korsman | Spatial variability in surface sediment composition characterised by near-infrared (NIR) reflectance spectroscopy[END_REF][START_REF] Rippey | The accuracy of methods used to estimate the whole-lake accumulation rate of organic carbon, major cations, phosphorus and heavy metals in sediment[END_REF]. In the rare occasion where a large number of samples (50) have been used [START_REF] Rada | Whole-lake burdens and spatial-distribution of mercury in Wisconsin seepage lakes[END_REF], the selected study sites were small lakes with simple basin morphometry, something that might oversimplify the observed patterns. In most studies there is also a tendency to focus on the organic fraction of the sediment, and thus, neglecting the inorganic fraction, which can be of large importance in determining, for example, the Hg bioavailability [START_REF] Gabriel | Principal biogeochemical factors affecting the speciation and transport of mercury through the terrestrial environment[END_REF].

[6] In the present study we aimed to take one step back and revisit fundamental questions of how the solid phase sediment composition -and especially total-Hg and methyl-Hgvaries spatially across a complex lake basin. Our specific research questions were: (i) How variable is the sediment composition within a single lake? (ii) What are the distribution patterns for total-Hg and methyl-Hg across the lake basin? (iii) Can the distribution of total-Hg and methyl-Hg be linked to the physical (i.e., bulk density and water depth) and geochemical properties (concentrations for 28 major and trace elements) of the sediment material, and if so, what are the most important links?

[7] In order to achieve our objective we collected surface sediment samples across the whole basin of the lake Stor-Strömsjön, a small boreal lake with two basins located in northern Sweden. We measured total-and methyl-Hg and characterized the geochemical composition of the sediment matrix using elemental analysis, wavelength dispersive X-ray florescence spectrometry (WD-XRF) and Fourier-transform infrared spectrometry (FTIRS). Finally the Hg data were coupled with the geochemical data and subjected to exploratory multivariate statistics (cluster analysis and principal component analysis (PCA)). These statistical tools help us to see beyond individual elements and thereby treat the sediment matrix as a complex unit that can be related, statistically, to the total-Hg and methyl-Hg distributions.

Materials and Methods

Study Site

[8] Stor-Strömsjön (64 07′N, 19 14′E, 209 m.a.s.l) is a relatively small, dimictic lake situated in northern Sweden, approximately 60 km NW of Umeå. Mean annual air temperature for the area is $3 C (SMHI). The area consists of old, slow-weathering metamorphic bedrock covered with poorly sorted glacial till (www.sgu.se). The catchment area is 93 km 2 and consists of a mixture of managed coniferous forest, mires and lakes. Human activities in the catchment consist of forestry, with a number of small gravel roads crossing through the area, and some small settlements, which include smaller parcels of agricultural land (Figure 1).

[9] Stor-Strömsjön has a lake area of 3.5 km 2 , which is divided into two main basins: a smaller and shallower eastern basin (1.1 km 2 ; maximum depth 4.2 m) and a larger and deeper western basin (2.4 km 2 ). The western basin can be sub-divided further into two sub-basins, one larger section in the northern end (maximum depth 8.2 m), and a smaller section in the southern end (maximum depth 11.8 m), hereafter referred to as the northwestern and southwestern basins, respectively. While smaller streams and ditches feed into the eastern and southwestern basins, the main inlet, which drains $64% of the lake's catchment, feeds into the northwestern basin. The only outlet is located on the southeastern end of the eastern basin and it has a small dam that has raised the water level of the lake by about 0.5 m.

Sampling

[10] Surface sediment samples from Stor-Strömsjön were collected in January 2009. A Geodolite total station was used to mark a 200 by 200 m grid on the lake ice, resulting in 85 potential sample locations. Two locations were removed because the sediment was too compact for sampling (F2 and G13), thus yielding a total of 83 sample locations. Water depth was recorded for each location, and the slope at each location was thereafter modeled using ArcGIS (www.esri.com). At each location the top 2 cm of the sediment was sampled using a HTH-gravity corer [START_REF] Renberg | The HTH sediment corer[END_REF]. A 0-2 cm sample provides enough material (2 g) for the analyses used, but at the same time it restricts the period during which the sediment has accumulated as much as possible. This is important for two reasons: first, a thin sample reduces the effect of differences in sediment accumulation rate across the lake basin, and second, a thin sample represents sediments in close connection with the water column that are more relevant for biogeochemical cycling of elements. Even so it is necessary to keep in mind that the top 2 cm of the sediment represent different time spans in different locations, and thus, it might be more or less affected by temporal changes in the Hg-loading. The samples were transferred in the field into new polypropylene boxes with polyethylene lids. In the lab all samples were weighed, freeze-dried, re-weighed to calculate dry bulk density (B.D.), and then homogenized and powdered prior to further analysis. The samples were stored in a dark freezer (À20 C) both before and after freeze drying to minimize any risk of organic matter degradation that could affect the FTIRS analysis.

Analysis 2.3.1. WD-XRF

[11] A Bruker Tiger S8, WD-XRF analyzer was used to determine the elemental concentrations of selected major (Na, Mg, Al, Si, K, Ca, P, S, Mn, Fe) and trace elements (Cl, Sc, Ti, V, Cr, As, Br, Y, Zr, Ni, Cu, Zn, Rb, Sr, Ba, Pb) in all 83 samples. About 2 g of powdered sediment material was analyzed and elemental concentrations were calculated using a calibration method specifically designed for lake sediments (modified from De Vleeschouwer et al. [2011]). Lower limits of detection (LLD) range from a few ppm for most trace elements to a few tens of ppm for major elements. Accuracy -assessed using 10 different CRM'swas within AE10% for K, Ca, Al, Si, Ti, Mn, Rb, Sc, Cl, Y, Cu, Zn, Ni and V, 10-17% for Na, Fe, Ba, Sr, Zr, S, As and Pb, and above 20% for Mg, Br, P and Cr. However, for most elements the reproducibility for replicate samples (n = 3) is within AE5%, with the exception of Sc, Br, As and Cr, which have reproducibilities of AE8, 10, 36 and 12%, respectively. Values for LLD, accuracy and reproducibility are provided in the auxiliary material.1 2.3.2. Carbon and Nitrogen

[12] C and N concentrations were determined simultaneously using a Perkin Elmer 2400 Series II CHNS/O-analyzer (operated in CHN-mode). The accuracy of reference samples (n = 28) was within AE10% and sample replicates (n = 24) were always within AE10% of each other. The C and N concentrations are reported as percent of sediment dry mass, and due to the non-calcareous bedrock in the area, total C can be considered to be equal to organic C. This assumption was also supported by the FTIRS spectra, where no peaks appear in spectral regions known to be associated with carbonates [START_REF] Rosén | Fourier transform infrared spectroscopy, a new method for rapid determination of total organic and inorganic carbon and biogenic silica concentration in lake sediments[END_REF].

FTIRS

[13] As a complement to the more traditional WD-XRF and elemental analysis, we also subjected the samples to FTIRS analysis. FTIRS uses the mid-infrared region and is sensitive to both inorganic constituents (primarily Fe, Mn, Si, carbonates and clay minerals), as well as, organic matter [START_REF] Belzile | Characterization of humic substances extracted from Canadian lake sediments[END_REF][START_REF] Farmer | The Infrared Spectra of Minerals[END_REF]. In this study FTIRS is primarily used to infer biogenic silica (BSi) and to assess whether there are any differences in organic matter quality between different parts of the lake, however, it is also used as supportive data for the data obtained with WD-XRF analysis.

[14] FTIRS analysis followed the procedure of [START_REF] Vogel | Fourier transform infrared spectroscopy, a new cost-effective tool for quantitative analysis of biogeochemical properties in long sediment records[END_REF]. In brief: The sediment was mixed with potassium bromide (KBr, Spectroscopic grade, Fisher Scientific), and FTIR spectra were obtained under vacuum using a Bruker IFS 66v/S FTIR spectrometer (Bruker Optik GmbH, Germany) equipped with a diffuse reflectance accessory (Harrick Inc., USA). FTIR spectra were obtained between 400 and 3750 cm À1 . All spectra were baseline corrected using two points, i.e., 2200-2210 and 3750 cm À1 in order to reduce the effect of possible instrumental drift.

FTIRS Inferred Biogenic Silica

[15] Biogenic silica (BSi) has a different FTIRS signature as compared to geogenic Si [START_REF] Gendron-Badou | Spectroscopic characterization of biogenic silica[END_REF], and [START_REF] Rosén | Universally applicable model for the quantitative determination of lake sediment composition using Fourier transform infrared spectroscopy[END_REF] have developed a method for inferring BSi from the FTIR spectra. In brief: The FTIR spectra of 816 sediment samples with known BSi concentrations (ranging from 0 to 55% as determined according to the method of [START_REF] Müller | An automated leaching method for the determination of opal in sediments and particulate matter[END_REF]). Using the software SIMCA-P 10.0 (Umetrics AB, SE-907 19 Umea , Sweden), a 4-component PLS model including spectral regions with known molecular vibrations of BSi (435-480, 790-830 and 1050-1280 cm À1 ) was developed [START_REF] Gendron-Badou | Spectroscopic characterization of biogenic silica[END_REF]. The model between the conventionally measured BSi and FTIRS-inferred BSi shows a cross-validated predictive power (R cv ) of 0.93.

Mercury

[16] Total-Hg concentrations were determined using a Leco AMA 254-analyzer (thermal decomposition-atomic absorption spectrometer). Calibration was based on analyses of different weights (20-200 mg) of MESS-3 (National Resources Canada, certified value 91 AE 9 ng g À1 ). CRMs were included along with the samples, and were within their respective certified ranges and all sample replicates (n = 23) were within AE10%.

Methylmercury

[17] Based on the WD-XRF results, a sub-set of 35 samples was selected for methyl-Hg analysis. In the selection procedure the aim was to retain as much as possible of the variability in the entire 83-sample data set, and also to have a reasonable spatial coverage within the lake basin. Methyl-Hg was analyzed according to the isotope dilution method described in [START_REF] Lambertsson | Applications of enriched stable isotope tracers in combination with isotope dilution GC-ICP-MS to study mercury species transformation in sea sediments during in situ ethylation and determination[END_REF]. The method precision was 5% RSD, based on replicate sample analyses (n = 3), and the method detection limit was 0.03 ng MeHg g À1 , calculated as 3-times the standard deviation (3s) of the blank.

Statistical Analysis

[18] All statistical analyses of the Stor-Strömsjön data were done using the SPSS software package PASW, version 18.0 (www.spss.com). Bi-variate correlation coefficients (r) were calculated as 2-tailed Pearson correlations and only correlations significant at the 0.05-level or above are reported. To distinguish bi-variate correlations from PC-loadings, which are reported as percent of variance explained, the correlation coefficients are given as r 2 -values. Due to a non-normally distributed sample population within clusters, the statistical significance of any differences in Hg between clusters was evaluated using an independent-sample Kruskal-Wallis test with the significance level set to 0.05. Prior to PCA and cluster analysis all data (also the PC FTIR scores) were converted to Z-scores (with average = 0 and variance = 1).

Cluster Analysis

[19] The observations (sample locations) are grouped into distinct clusters based on their geochemical composition. This makes it easier to identify the general patterns in sediment distribution in the lake, and to look beyond individual sample locations and individual variables. A geochemical data set, consisting of the WD-XRF data and the C, N, inferred BSi and bulk density, was subjected to a hierarchical agglomerative cluster analysis using Wards linkages [START_REF] Ward | Hierarchical grouping to optimize an objective function[END_REF] based on squared Euclidean distances; a solution of 5 clusters was selected to represent the data.

Principal Component Analysis

[20] FTIRS produce a large quantity of data -1738 spectral absorbance pointsfor each sample and because our intention is to use the FTIRS data together with the WD-XRF data in subsequent principal component analysis (PCA), there is a need to reduce the number of variables (otherwise the FTIRS data will totally dominate the PCA), but still retain as much as possible of the information regarding the sediment composition. This was achieved by first subjecting the FTIRS data alone to a PCA (PCA FTIR ). In this initial PCA we extracted variables with eigenvalues > 10, and used a Varimax rotated solution, which resulted in the extraction of six principal components (PC FTIR ) explaining 98% of the total variance.

[21] The scores of the six PC FTIR were then used as variables, together with the WD-XRF elemental data, water depth, basin slope, inferred BSi, C, N, total-Hg and methyl-Hg data, in two separate PCAs. The first was done using all 83 sample locations, but excluding methyl-Hg as a variable (PCA All ), while the second included only the reduced set of 35 locations where both tot-Hg and methyl-Hg were measured (PCA Red ). In both PCAs all components with eigenvalues > 1 were extracted using a Varimax rotated solution [START_REF] Kaiser | The Varimax criterion for analytic rotation in factor analysis[END_REF]. This type of rotation reduces the number of components used to explain each variable and results in cleaner, and more easily interpreted, factor loadings. Factor loading coefficients (loadings) were calculated as regression coefficients, which is analogous to r in Pearson correlations. For convenience the loadings are reported as percent of variance explained, i.e., as squared loadings. For all PCs, variables with loadings less than 15% were considered to be insignificant with respect to that PC, and hence, they are not further discussed in relation to that particular PC.

Results and Discussion

Sediment Geochemistry

[22] If some shallow locations with very low organic content are excluded the sediments of Stor-Strömsjön are relatively rich in organic matter (8.2% C, excluding cluster-1 locations). However, when subjected to cluster analysis, five different types of sediments, with different geochemical characteristics, can be identified (Table 1 andFigure 2). Together with the statistical association between different elements identified by the PCA's, this allows for making an interpretation of the major factors affecting sediment distribution in Stor-Strömsjön.

The PCAs

[23] For the whole 83-sample data set (PCA All ) seven principal components (PC All 1-7) were extracted, which together explain 89% of the total variance in sediment composition (Figure 3). In the reduced 35-sample PCA (PCA Red ) six PCs were extracted (PC Red 1-6), also explaining 89% of the total variance (Figure 4). The general patterns in PCA All and PCA Red are very similar, especially for tot-Hg, which indicates that the 35-sample sub-set selected for methyl-Hg analysis is representative for the whole-lake data set (83 locations). Here follows a short description of the seven PC All . Because the first three PCs are basically the same for both PCA All and PCA Red , no difference is made between PC All 1-3 and PC Red 1-3. For the remaining four (or three) PCs, they will be introduced using the order in which they appear in PCA All , with a reference to the corresponding PC Red .

PC-1: Mineral Composition

[24] The first PC (explaining 33 and 34% of total variance for PC All and PC Red , respectively) separates the sediment matrix into two minerogenic fractions. Positive loadings represent high concentrations of elements generally associated with silicate minerals (Mg, Al, Rb, K, Ti, Sr, Zr, Si), and thus, it indicates samples rich in silicates. Negative loadings represent higher concentrations of Fe, As, Mn, Zn together with C and Br. This is consistent with negative PC-1 scores being indicative of samples rich in Fe-/Mn-precipitates, which are known to often contain trace metals and organic matter due to co-precipitation during their formation Listed are also the averages values for each of the variable for each of the 5 clusters defined in Figure 2. For the cluster averages: italics denote values above whole-lake average (>10% above); boldface denotes values close to whole-lake average (AE10%); and nonbold, nonitalic entries are values below whole-lake average (>10% below). [START_REF] Deer | An Introduction to the Rock-Forming Minerals[END_REF][START_REF] Laing | Trace metal behaviour in estuarine and riverine floodplain soils and sediments: A review[END_REF][START_REF] Guo | The influence of sediment redox chemistry on chemically active forms of arsenic, cadmium, chromium, and zinc in estuarine sediment[END_REF]. That PC FTIR -1 loads together with Fe, As, Mn and Zn and orthogonally to Si, is consistent with the FTIR spectra, where locations with negative PC FTIR -1 have high spectral intensities in regions important for Si (e.g., a large peak centered at 1000 cm À1 [START_REF] Ellerbrock | Characterization of soil organic matter from a sandy soil in relation to management practice using FT-IR spectroscopy[END_REF]).

PC-2: Organic Matter

[25] For the second PC (explaining 21% of total variance for both PC All and PC Red ), elements that are either a part of, Figure 3. Combined score-and loading-plots for PCs 1-6 depicting how different variables and samples relate to each other in the ordination space as defined by PCA All (PCA using all 83 samples, but excluding MeHg; PC-7 not shown; cf. Figure 4). In brief, PC-1 is related to mineral composition (Fe/Mn-precipitates versus silicate minerals), PC-2 is related to the amount of organic matter, PC-3 relates to the mineral matter grain size and PC All -6 is related to water depth (the actual process behind PC-4 and 5 is less clear). Total-Hg plots mainly on the positive side of PC 2 and 3 (indicated by squared boxes), indicating that high total-Hg concentrations can be linked to higher amounts of organic matter and/or fine-grained mineral matter. The five different clusters represent samples characterized by: (1) coarse-grained minerogenic sediments with very low organic content, (2) fine-grained minerogenic sediments with low organic matter content, (3) Fe/Mn-precipitates, (4) a composition close to the whole-lake average and (5) higher organic matter content. To make the figures more legible all variables with PC-loadings of less than 0.39 for both plotted PCs have been removed and overlapping elements have been moved to the edge of the plot with lines marking their actual location.

or are strongly associated with, organic matter have positive loadings (C, N, Cl, Br, S, BSi, Pb, Cu, Hg) together with PC FTIR -2. FTIR spectra from locations with positive PC FTIR -2 scores show high spectral intensities in regions important for organic matter and BSi (e.g., 1100, 1300-1750 and 2800-3000 cm À1 [Gendron- [START_REF] Gendron-Badou | Spectroscopic characterization of biogenic silica[END_REF][START_REF] Mecozzi | Carbohydrates proteins and lipids in fulvic and humic acids of sediments and its relationships with mucilaginous aggregates in the Italian seas[END_REF]]. On the negative side of PC-2 we find bulk density together with some of the major constituents of minerogenic material (Ca, Al). However, PC-2 is not merely a measure of the organic matter concentrations (e.g., some of the C is associated with PC-1); it also tends to separate differences in organic matter quality. This is indicated by the positive loading for BSiwhich mostly originates from algal material with high carbon quality, as compared to terrestrial organic matterand a negative bivariate correlation between the PC-2 scores and C:N-ratio (r 2 = 0.44). The C:Nratios can be seen as a proxy for organic matter quality with low C:N values (4-10, atomic ratio) indicating fresh organic matter mostly produced within the lake and high values (>20) representing more degraded organic matter of terrestrial origin [START_REF] Meyers | Sediment organic matter[END_REF]. That Si loads on the positive side of this component can be explained by BSi being a significant part of the total Si-pool in Stor-Strömsjön.

PC-3: Mineral Matter Grain Size

[26] In the third PC (explaining 12 and 13% of total variance for PC All and PC Red , respectively) there is a split between water depth, some trace elements (Ti, K, Rb, Hg, Cu, Ni, V) and Mg on the positive side, and bulk density on the negative. On the positive side of the loading plot we also find PC FTIR -3. We suggest that PC-3 separates between coarse-and fine grained minerogenic sediments, because elements known to be enriched in the fine-grained clay Figure 4. Combined score-and loading-plots for PCs 1-6 depicting how different variables and samples relate to each other in the ordination space as defined by PCA Red (PCA using only 35 samples, but all variables including MeHg; cf. Figure 3). In brief, PC-1 is related to mineral composition (Fe/Mn-precipitates versus silicate minerals), PC-2 is related to the amount of organic matter, PC-3 relates to the mineral matter grain size and PC Red -5 is related to water depth and possibly anoxia (the actual process behind PC-4 and 6 is less clear). Total-Hg plots mainly on the positive side of PC 2 and 3 (indicated by squared boxes), indicating that high total-Hg concentrations can be linked to higher amounts of organic matter and/or fine-grained mineral matter (cf. Figure 3). Methyl-Hg is mainly associated with PC Red -5, but also to some extent to PC-2 (indicated by ellipses). This indicates that the main factor controlling methyl-Hg is water depth and that organic matter content is of only limited importance. The five different clusters represent samples characterized by: (1) coarse-grained minerogenic sediments with very low organic content, (2) fine-grained minerogenic sediments with low organic matter content, (3) Fe/Mn-precipitates, (4) a composition close to the whole-lake average and (5) higher organic matter content. To make the figures more legible all variables with PC-loadings of less than 0.39 for both plotted PCs have been removed and overlapping elements have been moved to the edge of the plot with lines marking their actual location.

fraction, for example, Ti, K and Rb, [START_REF] Koinig | 9000 years of geochemical evolution of lithogenic major and trace elements in the sediment of an alpine lake-The role of climate, vegetation, and land-use history[END_REF][START_REF] Taboada | Particle-size fractionation of titanium and zirconium during weathering and pedogenesis of granitic rocks in NW Spain[END_REF], have positive loadings, while bulk density, which is favored by a coarser grain size, has a negative loading. That water depth plots on the positive side would be consistent with the general model for sediment focusing, where fine-grained material tends to accumulate in deeper waters [START_REF] Håkanson | Principals of Lake Sedimentology[END_REF]; and a finer grain size would give a larger negatively charged surface area, as compared to coarser-grained material, and thus a greater ability to adsorb trace metals such as Hg, Cu, Ni and V [START_REF] Stone | Distribution of lead, copper and zinc in size-fractionated river bed sediment in two agricultural catchments of southern Ontario, Canada[END_REF]. 3.1.1.4. PCs 4 to 7

[27] PC All -4 (PC Red -6; both accounting for 7% of total variance) is mainly driven by changes in Cr and Mn, which both have a strong positive loading (for PC Red 6 also PC FTIR -6 is a strong driver). For PC All -5 (PC Red -4; 6 and 7% of total variance, respectively), P and Zn are the main drivers. PC All -6 (PC Red -5; 5 and 7% of total variance, respectively) is driven by water depth, which has a positive loading, and PC FTIR -5, which has a negative loading. In PC Red -5 methyl-Hg also plots on the positive side, together with water depth and S. For PC All -7 (4% of total variance) the driving variables are PC FTIR -6, with a positive loading, and basin slope, with a negative loading. PC Red -6 represents a combination of PC All -4 and PC All -7, and has positive loadings for Cr, Mn and PC FTIR -6, and a negative loading for basin slope.

Spatial Patterns in the Sediment Geochemistry

[28] The surface sediments from shallow locations close to the major inlet in the northwestern basin and minor inlets in the southwestern basin (cluster 1) have higher than wholelake average concentrations of major elements (e.g., Na, Mg, K, Ca, Al, Si), higher bulk density and low trace metal and organic matter concentrations (Table 1 andFigure 2). Taken together with the positive PC-1 and negative PC-2 and PC-3 scores, this indicates coarser-grained minerogenic sediments mainly consisting of silicate minerals (Figures 3 and4). This shows, as would be expected, that the inletsparticularly the main inletare major sources for silicate minerals to Stor-Strömsjön, and that the lighter fraction is depleted in these shallow areas due to water action [START_REF] Håkanson | Principals of Lake Sedimentology[END_REF].

[29] In the deeper areas of the northwestern basin (cluster 2) the surface sediment has similar concentrations for major elements as compared to cluster 1. Hence, cluster-2 locations plot on the positive side of PC-1 and negative side of PC-2, indicating that they consist of the same type of silicate minerals as cluster 1 (Table 1 andFigures 234). However, the surface sediments within cluster 2 have positive PC-3 scores and are enriched in several trace metals (i.e., Hg, Cu, Pb, Ni, V), which suggests a more fine-grained material. This would comply with a transport of fine-grained material from shallower to deeper areas of the northwestern basin [START_REF] Håkanson | Principals of Lake Sedimentology[END_REF]. That fine-grained minerogenic rich sediment (cluster 2) is mainly confined to the deep areas of the northwestern basinas opposed to the more organic rich sediments found in the southwestern (cluster 4) and eastern (cluster 5) basinssupports the idea that the main inlet, which drains 64% of the lake's catchment, is the main supplier of silicate mineral material to the lake.

[30] Along the shorelines of the two western basins (cluster 3) the surface sediments are generally rich in Fe and Mn, and have negative PC-1 and PC-2 scores and plot in the middle of PC-3 (Table 1 andFigures 234). This indicates a material high in Fe/Mn-minerals, which also is consistent with the presence of visually identifiable grains of Fe/Mnprecipitates in these samples. That the majority of the cluster-3 locations are found at water depths of 2.4-4 m suggests that the redox conditions in this depth zonefor example, during summer/winter stratificationfavor the formation of Fe and Mn precipitates [START_REF] Stumm | Aquatic Chemistry-Chemical Equilibria and Rates in Natural Waters[END_REF]. However, it is also possible that the distribution is linked to inflowing deep groundwater rich in reduced Fe-and Mn-species in these areas [START_REF] Stumm | Aquatic Chemistry-Chemical Equilibria and Rates in Natural Waters[END_REF].

[31] Surface sediments from locations found in the deepest part of the lake, i.e., in the southwestern basin and in the southern end of the eastern basin (cluster 4), have a composition close to the whole-lake average for most elements, and consequently plots in the middle of PC-1, PC-2 and PC-3 (Table 1 andFigures 234). A lower relative amount of mineral material in the deep areas of the southwestern, as opposed to the northwestern, basin is consistent with the assumption that silicate mineral material mainly is supplied through the main inlet. For the eastern basin there is a higher input of mineral material to the southern part, as opposed to the northern part, of the basin as a result of water currents transporting mineral material from the western basin toward the outlet.

[32] Locations with surface sediments with higher than average concentrations of organic matter (cluster 5) are generally found in the northern half of the eastern basin, and plot to the positive side of PC-2 and in the middle of PC-1 and PC-3 (Table 1 andFigures 2-4). In-lake production, either pelagic or benthic, is likely an important source of this organic matter given the high concentration of BSi, probably from diatoms. However, the C:N ratio is higher (11-14) than for pure algal material (4-10), and even if differences in nitrogen cycling can affect the C:N-ratio, this indicates that there is also a contribution of terrestrial organic material, presumably through small inlets draining mire areas located north of and along the shores of the eastern basin [START_REF] Meyers | Lacustrine organic geochemistry-An overview of indicators of organic matter sources and diagenesis in lake sediments[END_REF].

[33] Taken separately, each of the three sub-basins complies relatively well to the generalized model of sediment focusing [START_REF] Blais | The influence of lake morphometry on sediment focusing[END_REF][START_REF] Håkanson | Principals of Lake Sedimentology[END_REF]. However, on a whole-lake basis there is no correlation between, for example, water depth and organic matter. It is also clear that several locations deviate from the withinbasin patterns. This suggests that there is an influence of the catchment properties in the immediate vicinity of these locations [cf. [START_REF] Korsman | Spatial variability in surface sediment composition characterised by near-infrared (NIR) reflectance spectroscopy[END_REF]. For example, a number of small inlets along the western shore of the northwestern basin, a couple of small mires located near the shore of the southwestern basin and agricultural fields located just east of the eastern basin.

Total-and Methyl-Hg

[34] For the 83 locations total-Hg concentrations range from 12 to 140 ng g À1 , with a whole-lake average of 96 ng g À1 (Table 2 andFigure 5). Total-Hg inventories (i.e., the amount of total-Hg in the top 2 cm per unit area) in the surface sediment also show considerable variation, between 18 and 81 mg m À2 , with a whole-lake average of 39 mg m À2 . In the sub-set of 35 locations, methyl-Hg concentrations vary between 0.07 and 2.45 ng g À1 (whole-lake average, 0.70 ng g À1 ; Table 2 andFigure 5), and methyl-Hg inventories range from 0.11 to 1.37 mg m À2 (whole-lake average, 0.30 mg m À2 ). The methyl-Hg:total-Hg ratio (MeHg:TotHg) varies between 0.2 to 3.3%, with an average ratio of 0.79% for the whole lake.

Total-Hg in Surface Sediments of Stor-Strömsjön

[35] In both PCA All and PCA Red two principal components, together explaining $70% of the total-Hg variance, are of almost equal importance for explaining total-Hg concentrations in surface sediments from Stor-Strömsjön (Figures 3 and4). One PC is associated to organic matter (PC-2; 34 and 43% for PCA All and PCA Red , respectively) and the other to fine-grained mineral matter (PC-3; 36 and 33% for PCA All and PCA Red , respectively). That such a large part of total-Hg is statistically associated with mineral matter is interesting, and partly contradicts the general assumption that total-Hg in the boreal landscape is almost exclusively controlled by organic matter [START_REF] Meili | The coupling of mercury and organic matter in the biogeochemical cycle-Towards a mechanistic model for the boreal forest zone[END_REF][START_REF] Ullrich | Mercury in the aquatic environment: A review of factors affecting methylation[END_REF]. Even so, the capacity for mineral particles to adsorb Hg is well documented [START_REF] Gabriel | Principal biogeochemical factors affecting the speciation and transport of mercury through the terrestrial environment[END_REF][START_REF] Schuster | The behavior of mercury in the soil with special emphasis on complexation and adsorption processes-A review of the literature[END_REF], especially if the mineral grains are coated with organic substances, for example, Averages are also calculated for each of the five clusters defined in Figure 2. For the cluster averages: italics denote values above whole-lake average (>10% above); boldface denotes values close to whole-lake average (AE10%); and nonbold, nonitalic entries are values below whole-lake average (>10% below).

b Clusters with average values that differ from all other clusters with statistical significance.

Figure 5. The spatial distribution of total-Hg concentration, total-Hg inventories, methyl-Hg concentration, methyl-Hg inventories and methyl-Hg:total-Hg ratio. Ranges for the symbol classes are defined using natural breaks in the sample distribution (as defined by ArcGIS; www.esri.com). To make it easier to relate the spatial distribution of the Hg data it has been overlain on the cluster map from Figure 2. Contour intervals are 4 m.

fulvic acids [START_REF] Bäckström | Effects of a fulvic acid on the adsorption of mercury and cadmium on goethite[END_REF]; and a strong connection between fine-grained mineral matter and total-Hg concentrations is reported from studies conducted in areas outside of the boreal region [START_REF] Roulet | Increase in mercury contamination recorded in lacustrine sediments following deforestation in the central Amazon[END_REF][START_REF] Yin | Mercury concentrations in stream and river water: An analytical framework applied to Minnesota and Wisconsin (USA)[END_REF].

[36] If the spatial patterns of total-Hg in surface sediments are studied based on the cluster analysis it is interesting to note that there are no significant differences in total-Hg concentrations between clusters 2, 4-5, even if the sediment composition is different (more minerogenic, intermediate and more organic, respectively). This suggests that, even though the two PCs that are equally important for explaining total-Hg on a whole-lake basis, they have a different relative importance in different parts of the lake. This can be assessed by looking at the bi-variate correlation between total-Hg versus C (a proxy for organic material) and total-Hg versus PC-3 scores (the grain-size related PC), which provide an assessment of the predictive power of C and PC-3, respectively, for total-Hg concentrations. If all sample locations are included in the correlation analysis, the regression line has an r 2 of 0.44 and 0.35 for total-Hg versus C and total-Hg versus PC-3, respectively. However, if the cluster-2 locations sediments rich in fine-grained mineral materialare removed from the total-Hg versus C correlation then the r 2 increases from 0.44 to 0.58. And similarly, if the cluster-5 locationssediments rich in organic matterare removed from the total-Hg versus PC-3 correlation the predictive power of fine-grained mineral material increases from 0.35 to 0.55. These spatial patterns support the notion from the PCA that both organic matter and fine-grained mineral matter are important transport vectors for total-Hg into and within Stor-Strömsjön. The main transport route for mineral bound total-Hg likely is through the main northern inlet, whereas the origin of the organically bound total-Hg can be linked to two separate, but intertwined, processes. First, total-Hg can be transported to the lake along with terrestrial organic material from the catchment, both through the main inlet but maybe more importantly from the smaller inlets draining mire areas in close proximity to the lake (e.g., around the northern side of the eastern basin). Second, Hg atmospherically deposited directly on the lake surface can bind and settle with organic matter in the water column [START_REF] Hines | Mercury inputs and outputs at a small lake in northern Minnesota[END_REF][START_REF] Meili | The coupling of mercury and organic matter in the biogeochemical cycle-Towards a mechanistic model for the boreal forest zone[END_REF][START_REF] Outridge | Evidence for control of mercury accumulation rates in Canadian high arctic lake sediments by variations of aquatic primary productivity[END_REF].

[37] For total-Hg concentrations organic matter and finegrained mineral material are of equal importance, but if the total-Hg inventories (i.e., the amount of tot-Hg in the upper most 2 cm) is considered, the most important transport route for total-Hg is likely together with fine-grained mineral material entering the lake through the main inlet. This idea is supported by cluster 2 having significantly higher total-Hg inventories as compared to other clusters (Table 2). That an increased catchment to lake area ratio increases the total-Hg loading is well known from other studies, for example, [START_REF] Swain | Increasing rates of atmospheric mercury deposition in midcontinental North America[END_REF], but it is interesting that the difference between the eastern and northwestern basinswhere the northwestern basin has a considerably larger catchment to lake area ratiomight be linked to differences in the sediments geochemical composition.

[38] Total-Hg in hypolimnic waters and sediments has previously been linked to other elements, for example, Fe and Mn [START_REF] Chadwick | Influences of iron, manganese, and dissolved organic carbon on the hypolimnetic cycling of amended mercury[END_REF][START_REF] Regnell | Effects of anoxia and sulfide on concentrations of total and methyl mercury in sediment and water in two Hg-polluted lakes[END_REF]. However, in the case of Stor-Strömsjön no such links could be made, possibly due to the large differences in mineralogical composition between sampling locationsfor example, very high concentrations of Fe/Mn-precipitates in some locationswhich might cloud any such relationship present in deeper, more anoxic waters.

Sediment Distribution of Methyl-Hg

[39] Whereas total-Hg concentrations are largely controlled by a combination of organic and fine-grained mineral matter (PC-2 and PC-3), most of the variance in methyl-Hg concentrations (55%) is explained by PC Red -5 (PC All -6). Other variables plotting on PC Red -5 are water depth (39%), S (16%), and PC FTIR -5 (67%). A smaller part, 20%, of the methyl-Hg concentration is associated with PC-2 (organic matter), which also is important for C (66%), total-Hg (45%), S (52%), BSi (44%) and PC FTIR -2 (91%). However, even if both C and total-Hg are correlated to the same PC as methyl-Hg they are not necessarily directly correlated to methyl-Hg. If the variance in methyl-Hg explained by PC Red -5 is removed (i.e., calculating the residuals; Res-MeHg), there is no bivariate correlation between either Res-MeHg and C or total-Hg (p-values of 0.068 and 0.151, respectively). Both for C and total-Hg there are examples of studies in support of both a poor and good correlation between methyl-Hg and these variables; [START_REF] Lambertsson | Organic material: The primary control on mercury methylation and ambient methyl mercury concentrations in estuarine sediments[END_REF] and [START_REF] Ethier | Mercury empirical relationships in sediments from three Ontario lakes[END_REF] found good correlations between methyl-Hg and C, while, for example, [START_REF] He | Horizontal and vertical variability of mercury species in pore water and sediments in small lakes in Ontario[END_REF] found the opposite pattern. For methyl-Hg and total-Hg, [START_REF] Ethier | Mercury empirical relationships in sediments from three Ontario lakes[END_REF] also found a good correlation with total-Hg, but [START_REF] Benoit | Geochemical and biological controls over methylmercury production and degradation in aquatic ecosystems[END_REF] reported a poor correlation.

[40] That methyl-Hg is mostly associated to PC Red -5, which only accounts for 5% of the total variance in sediment composition, indicates that sediment geochemistry is not critically important for the spatial distribution of methyl-Hg in surface sediments from Stor-Strömsjön. This results in a much larger spatial variability in methyl-Hg as compared to total-Hg (Figure 5). It is also reflected by the fact that there are no significant differences in methyl-Hg concentrations between clusters 1-2 and 4-5 (Table 2), and that the four highest methyl-Hg concentrations are found in locations with contrasting surface sediment types; i.e., deep locations rich in both finegrained mineral matter and organic matter (2.45 and 1.65 ng g À1 ; cluster 4), a shallow location with organic rich sediments (1.20 ng g À1 ; cluster 5) and a shallow location with coarsegrained minerogenic sediments (1.15 ng g À1 ; cluster 1).

[41] Based on these results together with knowledge on the behavior of methyl-Hg and S, it is possible to at least speculate on the controlling mechanisms behind the patterns in methyl-Hg concentrations observed in surface sediments from Stor-Strömsjön. Most of the variance in methyl-Hg concentrations is linked to water depthwhich in a dimictic lake could be seen as a proxy for anoxia, which has been shown to be important for methyl-Hg [START_REF] Benoit | Geochemical and biological controls over methylmercury production and degradation in aquatic ecosystems[END_REF][START_REF] Grigal | Mercury sequestration in forests and peatlands: A review[END_REF][START_REF] He | Horizontal and vertical variability of mercury species in pore water and sediments in small lakes in Ontario[END_REF] and a fraction (16%) of the Si.e., an element that is important for methylating sulfur reducing bacteria (SRB) [START_REF] Benoit | Geochemical and biological controls over methylmercury production and degradation in aquatic ecosystems[END_REF][START_REF] Ullrich | Mercury in the aquatic environment: A review of factors affecting methylation[END_REF]. This suggests that in situ production of methyl-Hg, possibly by SRBs stimulated by anoxia and S, are important for the distribution of methyl-Hg concentrations in Stor-Strömsjön, at least on a whole-lake basis. The co-variation between S and methyl-Hg could also arise simply because S-groups are of importance for the binding of methyl-Hg to organic matter [START_REF] Qian | Bonding of methyl mercury to reduced sulfur groups in soil and stream organic matter as determined by X-ray absorption spectroscopy and binding affinity studies[END_REF]. However, if this were the case we would expect that organic matter also would load on PC-5 (or that a higher portion of the methyl-Hg variation would be explained by PC-2). In fact most of the S (52%), which presumably represents organically bound S, loads on PC-2 (i.e., organic matter), but this PC still only accounts for a smaller part of the methyl-Hg (20%).

[42] Even if PC Red -5 is the most important PC for methyl-Hg concentrations, it still only accounts for 55% of the total variance. A part of the remaining variance (20%) is accounted for by PC-2 and organic matter. This association could be due to at least two different processes. One possibility is that the association between methyl-Hg and PC-2 arises from a transport of methyl-Hg from the catchment to the lake, something that is known to be important in some systems [START_REF] Branfireun | Speciation and transport of newly deposited mercury in a boreal forest wetland: A stable mercury isotope approach[END_REF]. However, higher PC-2 scores do not only indicate more organic matter, they also indicate a lower C:N ratio, and thus a higher content of algal material which is a better substrate for bacterial growth as compared to terrestrial organic matter [START_REF] Gudasz | Constrained microbial processing of allochthonous organic carbon in boreal lake sediments[END_REF]. That methyl-Hg is actually more associated with the quality of carbon than the quantity is also reflected as a weak bivariate correlation between residual methyl-Hg (after removing the explanatory power of PC Red -5) and BSi (r 2 = 0.20). Hence, the other possible explanationand maybe the more likely one based on this data setis that, in addition to the methylation pathway stimulated by water depth (anoxia) and S, there is a second methylation pathway, which is controlled by organic matter quality. A recent paper by [START_REF] Hamelin | Principal methylators of mercury in Lake Periphyton[END_REF] also stresses that not only SRBs, but also other microorganisms such as methanogens in shallow waters might be important for the methylation of Hg.

[43] However, even if both PC Red -5 and PC-2 are considered this still leaves about 25% of the total variance in methyl-Hg not accounted for. This would imply that there are additional factors, and additional variables, that are important for methyl-Hg distribution in Stor-Strömsjön. These might include the supply of terrestrially produced methyl-Hg, sulfur/Hg-speciation, pore water concentrations of Hg and S, groundwater inflow, microbial activity, demethylation rates and temperature.

[44] As with total-Hg, some spatial patterns emerge when methyl-Hg is viewed in the light of the two most important PCs. The predictability of PC Red -5 is reasonably good throughout the lake, but it tends to underestimate the methyl-Hg concentrations in the organic rich sediments of cluster 5 and the two deepest locations within cluster 4 (i.e., the two locations in the western basin with the highest scores on PC-2). If the predictability of PC-2 is assessed in relation to the Res-MeHg (with the effects of water depth and S removed), the performance is better for the cluster-5 locations, but there is still a tendency to underestimate methyl-Hg in the two deepest locations and in locations within cluster 1. This suggests that the PC Red -5, and any process behind this PC, is valid throughout the lake, and that PC-2, and the possible link between methyl-Hg and organic matter quality, is most valid for the sediments of cluster 5. That a part of the unaccounted variance is constrained to shallow locations with highly minerogenic sediments indicates that variables other than those considered in this study are of importance in that particular type of sediment.

[45] For MeHg:THg ratios and methyl-Hg inventories, as compared to concentrations, the within cluster heterogeneity is even larger and none of the clusters are significantly different from the others. Even so, the highest MeHg:THg ratios and methyl-Hg inventory were found within cluster 1, which also has the highest within-cluster average. All other clusters are close to or slightly below the whole-lake average for methyl-Hg inventories (Table 2 andFigure 5). That the highest inventory of methyl-Hg, and MeHg:TotHg, is found in a shallow location with coarse-grained minerogenic sediments is interesting, and is consistent with what [START_REF] Ethier | Mercury empirical relationships in sediments from three Ontario lakes[END_REF] found in three North American lakes. A large amount of methyl-Hg present in surface sediments in the shallow oxic zone might be an important factor to consider when assessing the bioavailability of methyl-Hg.

Conclusions

[46] In Stor-Strömsjön $70% of the spatial variance in total-Hg concentrations in surface sediments is controlled by the sediment composition, either by the organic matter content or the amount of fine-grained minerogenic material. This emphasizes that not only organic matter needs to be considered when assessing spatial as well as temporal changes in total-Hg concentration in sediments from boreal lakes. For methyl-Hg our interpretation is that in situ methylation is of greatest importance for the observed patterns; however, further studies are needed to better constrain the underlying processes. The other take home message from this study is that there can be significant spatial variability in both concentrations and inventories, especially for methyl-Hg. This is important to recognize both when sediments are used to calculate wholelake burdens of Hg, and when using concentrations or inventories from single sediment cores to compare different lakes.

[47] Our study also shows that the combination of WD-XRF and FTIRStwo rapid, data-rich analytical techniqueshas a large potential for distinguishing different sediment quality, and that these techniques can help to improve our understanding of how elements are distributed in lake basins and what underlying mechanisms account for these distributions. With further development of, for example, calibration models such as for BSi, the FTIRS technique can prove to become even more valuable for biogeochemical studies. Being fast and relatively inexpensive both WD-XRF and FTIRS can also be used to identify interesting patterns in large sample sets, and thereby provide the means to reduce the number of samples needed to be analyzed with more expensive and time consuming techniques, for example, potential methylation, microbial activity, which are needed to more fully understand the behavior of total-and methyl-Hg.

[48] Acknowledgments. This study was supported financially by the Swedish Society for Anthropology and Geography (SSAG) through a stipend awarded to the first author. Financial support was also provided by the Faculty of Science and Technology at Umeå University, the Swedish Research Council, FORMAS and the Kempe Foundations. We also thank the students from the Environmental Change Assessment-course in 2009 (Umeå University) for their help during field sampling and initial preparation of the samples, Johanna Engström (Landscape Ecology-group, Umeå University) for her help with the Geodolite total station and Per Person (Department of Chemistry, Umeå University) for the use of their FTIRS instrument.

Figure 1 .

 1 Figure 1. Maps of Stor-Strömsjön and its surroundings. (a) The location of Stor-Strömsjön in Scandinavia. (b) A schematic map over the Stor-Strömsjön catchment with lakes, mires and larger streams indicated. The wind rose indicates the predominant wind directions. (c) An aerial photograph of the lakes immediate surrounding combined with a bathymetric map over the lake where the 83 sample locations have been marked. Contour intervals in the bathymetric map are 2 m.

Figure 2 .

 2 Figure 2. A dendrogram showing the relationship between the five clusters identified in the cluster analysis (left), and the spatial distribution of the different clusters across the lake basin (right). Contour intervals are 4 m.

Table 1 .

 1 Minimum, Maximum and Whole-Lake Average Concentrations for All Variables Analyzed With WD-XRF, FTIRS-Inferred BSi, Bulk Density (B.D.), C, N, Water Depth (W.D.) and Slope

				Whole-Lake Values			Cluster Average a
	Variable	Unit	Min	Max	Average	1 (n = 9)	2 (n = 10)	3 (n = 22)	4 (n = 22)	5 (n = 20)
	Na	ppm	1716	10714	3777	6741	5383	2894	3921	2452
	Mg	ppm	2702	9303	6058	6827	8626	5239	6722	4598
	K	ppm	3234	8837	5887	7556	8186	5017	6380	4402
	Ca	ppm	4569	13362	7550	10759	8727	6977	7629	6059
	Al	%	1	2.7	1.8	2.2	2.5	1.6	2	1.4
	Si	%	15.6	30.8	23.2	26.6	24.8	18.2	23.6	26
	Ti	ppm	955	3440	2310	2993	3197	1925	2519	1754
	Fe	%	1	11.4	5.3	2.4	3.1	8.7	4.6	4.8
	Mn	ppm	445	59542	9586	2602	2359	20760	6774	7142
	Ba	ppm	295	1995	1210	1663	1710	1198	1122	864
	Rb	ppm	31	120	70	91	107	53	77	55
	Sr	ppm	49	207	94	157	124	77	96	70
	Sc	ppm	4	11	7	9	1 0	6	8	6
	Y	ppm	13	28	21	25	25	18	22	20
	Zr	ppm	76	390	143	257	183	114	143	106
	S	%	0.03	0.14	0.08	0.06	0.06	0.06	0.08	0.1
	P	ppm	75	1360	440	288	410	439	535	422
	Cl	ppm	57	158	102	90	93	94	104	119
	Br	ppm	3	30	15	8	11	15	15	21
	As	ppm	0	185	64	21	16	118	48	64
	Cu	ppm	4	18	12	8	16	9	14	14
	Zn	ppm	43	236	123	77	132	136	122	127
	Pb	ppm	0	36	23	20	26	17	22	29
	Ni	ppm	12	39	23	20	31	22	25	21
	V	ppm	36	79	58	57	74	54	64	48
	Cr	ppm	63	921	228	203	129	362	199	172
	BSi B.D.	% g cm À3	0.04 0.038	24.5 0.591	10.4 0.095	5 0.209	7.2 0.097	6.9 0.099	9.9 0.067	18.6 0.07
	C	%	1.6	11.9	8	5.6	6.8	7.5	8.4	9.7
	N	%	0.1	1	0.6	0.4	0.5	0.5	0.6	0.8
	W.D.	m	0.6	11.4	3.9	2.3	7.2	3.5	4.7	2.1
	Slope		0	17.4	1.0	0.8	0.3	2.2	0.5	0.8
	a

Table 2 .

 2 Minimum, Maximum and Whole-Lake Average for Total-Hg Concentration, Total-Hg Inventory, Methyl-Hg Concentration, Methyl-Hg Inventory and MeHg:TotHg Ratio

					All Sample Locations			Cluster Average a
		Unit	n	Min	Max	Average	1	2	3	4	5
	Tot-Hg Tot-Hg-Acc MeHg MeHg-Acc	ng g À1 mg m À2 ng g À1 mg m À2	83 83 35 35	11.7 18 0.07 0.11	137 81 2.45 1.37	95.9 39 0.70 0.30	46 b 38 0.72 0.67	109 53 b 0.75 0.32	87 b 36 0.42 0.19	107 35 0.92 0.29	109 38 0.71 0.25
	MeHg:Tot-Hg	%	35	0.20	3.3	0.79	1.6	0.7	0.5	0.8	0.7
	a

Auxiliary materials are available in the HTML. doi:10.1029/

2012JG001992.


Si Chen

Denis Efimov

Jean-Pierre Richard

Zongxia Jiao

Interval estimation for systems with unknown input delays and gains

Keywords: Interval observers, Quasi-monotone/Metzler/ cooperative systems, Time-delay systems, Uncertain input gains, Thermic application

The estimation problem of a system with unknown time-delay and unknown input gains is considered. The interval observation technique is applied in order to obtain guaranteed interval of the system state. The proposed approach can be applied to linear delay systems and nonlinear time-delay systems in the output canonical form. The theoretical results are supported by numerical simulation and demonstration on example of thermic application.

I. INTRODUCTION

An unknown time-delay and an unknown input gains in models of control systems arise due to many reasons in various real-world applications. The time-delay may be related to transport delays (like in chemical, hydraulic or pneumatic systems) or computational delays (e.g. in digital controllers or communication networks) [START_REF] Kruszewski | A switched system approach to exponential stabilization through communication network[END_REF]. The uncertain gains are related with possible nonlinearities and identification errors. The problem of observer design for nonlinear systems with delays and uncertainty is rather complex [START_REF] Sipahi | Stability and stabilization of systems with time delay limitations and opportunities[END_REF]. Especially the observer synthesis is problematical for the cases when the model of a nonlinear delayed system contains parametric and signal uncertainties, or when the delay is time-varying or uncertain [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], [START_REF] Briat | Design of LPV observers for LPV time-delay systems: an algebraic approach[END_REF], [START_REF] Califano | On the observer canonical form for Nonlinear Time-Delay Systems[END_REF], [START_REF] Darouach | Linear functional observers for systems with delays in state variables[END_REF], [START_REF] Germani | A state observer for nonlinear delay systems[END_REF], [START_REF] Sename | Observer-based H∞ control for timedelay systems: A new LMI solution[END_REF], [START_REF] Fattouh | Robust observer design for time-delay systems: a Riccati equation approach[END_REF], [START_REF] Seuret | Observer design for systems with nonsmall and unknown time-varying delay[END_REF], [START_REF] Zheng | On observation of time-delay systems with unknown inputs[END_REF], [START_REF] Gouaisbaut | Robust sliding mode control of nonlinear systems with delay: A design via polytopic formulation[END_REF]. An observer solution for this kind of problem is demanded, especially when the input gain of the system is also unknown.

Typically, in various applications the input delay and input gain are related. They both depend on the actuating system that may vary with time, environment or failures. An interval observer is expected to keep on providing a real-time estimation of the state variables in spite of these variations. Recent results [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], [START_REF] Polyakov | Output Stabilization of Time-Varying Input Delay Systems Using Interval Observation Technique[END_REF] have applied interval observers to the systems with unknown input delay and known parameters. The present work constitutes an extension to systems with unknown input delays and unknown (bounded) input gains.

The conventional observer in the absence of measurement noise and uncertainties has to converge to the exact value of the state of the estimated system (it gives a pointwise estimation of the state) [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF]. In opposite, the interval observers evaluate at each time instant a set of admissible values for the state, consistently with the measured output (i.e. they provide an interval estimation) [START_REF] Gouze | interval observers for unsertain biological systems[END_REF], [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF], [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF], which provides the guaranteed interval estimates of the system state in real-time. This property will simplify the control of transition processes with respect to system state. Such an interval estimation gives also a simultaneous estimation of the estimation error. An interval observer for time-delay systems can naturally deal with uncertain systems.

In some cases, the input gain has important impact in the design of observer which cannot be ignored. In this work, a new interval observer is presented, the case of time-varying uncertain delays is analyzed and the unknown input gains is additionally studied.

The paper is organized as follows. Some preliminaries are given is Section 2. The observer definition is given in Section 3, in the same section the observer design is performed. Last, in Section 4, an experimental example is considered: the results are applied to a thermic process, in which both transportation delay and electrothermic input gain depend on the external air flow.

II. NOTATION AND DEFINITIONS

In the rest of the paper, the following definitions will be used:

• R is the Euclidean space (R + = {τ ∈ R : τ ≥ 0}), C τ = C([-τ, 0], R) is the set of continuous maps from [-τ, 0] into R; C τ + = {y ∈ C τ : y(s) ∈ R + , s ∈ [-τ, 0]}; • x t is
→ R p the symbol ||u|| [t0,t1] denotes its L ∞ norm ||u|| [t0,t1] = ess sup{||u(t)||, t ∈ [t 0 , t 1 ]}, or simply ||u|| if t 1 = +∞,
the set of all such inputs u ∈ R p with the property ||u|| < ∞ will be denoted as L p ∞ ; • for a matrix A ∈ R n×n the vector of its eigenvalues is denoted as λ(A);

• E n ∈ R n is stated for a vector with unit elements, I n and 0 n denote the identity and zero matrices of dimension n × n respectively; • for two integers n ≤ N the symbol n, N denotes the sequence n, n + 1, . . . , N -1, N ;

• a R b corresponds to an elementwise relation R (a and b are vectors or matrices): for example a < b (vectors) means ∀i :

a i < b i ; for φ, ϕ ∈ C n τ the relation φ R ϕ has
to be understood elementwise for all domain of definition of the functions, i.e. φ(s) R ϕ(s) for all s ∈ [-τ, 0];

• denote µ [τ ,τ ] h(t) = inf s∈[τ ,τ ] h(t -s) and µ [τ ,τ ] h(t) = sup s∈[τ ,τ ] h(t -s) for a signal h : R → R.

A. Functional Differential Equation

A large number of processes can be modeled by a Functional Differential Equation (FDE):

ẋ(t) = f (t, x(t), x t , d), y(t) = h(t, x(t), x t , d), (1)
x t0 = ϕ ∈ C n τ , where t ∈ R is the time variable, d ∈ S d is either a vector or a function representing disturbances or parameter uncertainties of the system, S d ⊂ L q ∞ is a set of vectors or functions for which some bounds are usually supposed to be known, x(t) ∈ R n is a vector of internal variables, x t ∈ C n τ and τ ∈ R + is the maximal delay, y(t) ∈ R p is the output vector.

It is assumed that the system (1) has solutions (for example f satisfies Carathéodory conditions, see [START_REF] Hale | Theory of Functional Differential Equations[END_REF]) defined over a maximal interval denoted by I (1) (t 0 , ϕ) where t 0 is the initial time and ϕ is the initial function from C n τ .

B. Comparison/cooperative systems

Following the Wazewski's contribution [START_REF] Wazewski | Systèmes des Équations et des Inégalités Différentielles Ordinaires aux Seconds Membres Monotones et leurs Applications[END_REF], which is probably one of the most important in this field concerning differential inequalities and giving necessary and sufficient hypotheses ensuring that the solution of ẋ = f (t, x), with initial state x 0 at time t 0 and function f satisfying the inequality f (t, x) ≤ g(t, x) is overvalued by the solution of the so-called "comparison system" ż = g(t, z), with initial state z 0 ≥ x 0 at time t 0 , or, in other words, the conditions on function g that ensure x(t) ≤ z(t) for t ≥ t 0 . These results were extended to many different classes of dynamical systems [START_REF] Bitsoris | Principe de Comparaison et Stabilité des Systèmes Complexes[END_REF], [START_REF] Dambrine | Contribution à l'étude des systèmes à retards[END_REF], [START_REF] Matrosov | Vector Lyapunov Functions in the Analysis of Nonlinear Interconnected System[END_REF], [START_REF] Perruquetti | Connecting Wazewski's condition with Opposite of M-Matrix: Application to Constrained Stabilization[END_REF], [START_REF] Tokumaru | Macroscopic stability of interconnected systems[END_REF], [START_REF] Tchangani | Stability, attraction domains and ultimate boundedness for nonlinear neutral systems[END_REF]. Frequently these systems are also called monotone or cooperative [START_REF] Smith | Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems[END_REF]. Further in this subsection the exposition from [START_REF] Borne | Vector Lyapunov functions: nonlinear, time-varying, ordinary and functional differential equations[END_REF] will be adopted.

Focusing on two systems:

ẋ(t) = f (t, x(t), x t ), x(t) ∈ R n , (2)
ż(t) = g(t, z(t), z t ), z(t) ∈ R n , (3)
the solutions of (3) with initial condition ϕ 2 and of (2) with initial condition ϕ 1 will be denoted as z(t; t 0 , ϕ 2 ) and x(t; t 0 , ϕ 1 ) respectively.

Definition 1. The system (3) is said to be a comparison system of (2) over

Ω ⊂ C n τ if ∀(ϕ 1 , ϕ 2 ) ∈ Ω 2 : I = {t 0 }, I = I (2) (t 0 , ϕ 1 ) ∩ I (3) (t 0 , ϕ 2 ), ϕ 2 ≥ ϕ 1 =⇒ z(t; t 0 , ϕ 2 ) ≥ x(t; t 0 , ϕ 1 ) ∀t ∈ I.
Obviously, one can go beyond this concept to derive a qualitative analysis for positive solutions. For example, if z(t; t 0 , ϕ 2 ) ≥ x(t; t 0 , ϕ 1 ) ≥ 0 and if solution z(t) converges to zero so does x(t). A question naturally arises concerning the properties of the function g ensuring that (3) is a comparison system of (2) over Ω. For this, the following notion is required:

Definition 2. A functional g : R × R n ×C n τ → R n (t, x, y) → g(t, x, y) is quasi-monotone non-decreasing in x iff: ∀t ∈ R, ∀y ∈ C n τ , ∀(x, x ′ ) ∈ R n × R n ∀i ∈ 1, n : (x i = x ′ i ) ∧ (x ≤ x ′ ) ⇒ g i (t, x, y) ≤ g i (t, x ′ , y),
is non-decreasing in y iff:

∀t ∈ R, ∀x ∈ R n , ∀(y, y ′ ) ∈ C n τ × C n τ : y ≤ y ′ ⇒ g(t, x, y) ≤ g(t, x, y ′ ),
is mixed quasi-monotone non-decreasing in x, non-decreasing in y iff:

∀t ∈ R, ∀(x, x ′ ) ∈ R n × R n , ∀(y, y ′ ) ∈ C n τ × C n τ ∀i ∈ 1, n : (x i = x ′ i ) ∧ (x ≤ x ′ ) ∧ (y ≤ y ′ ) ⇒ (g i (t, x, y) ≤ g i (t, x ′ , y ′ )).
Remark 1. The latter definition is a special case of mixed quasimonotonicity given in [START_REF] Laksmikantham | Differential and integral inequalities,Vol I and II[END_REF]. More general versions also exist (see [START_REF] Bitsoris | Stability Analysis of Non Linear Dynamical Systems[END_REF], [START_REF] Habets | Attractivity Concepts and Vector Lyapunov Functions[END_REF]) and additional conditions are sometimes given (see [START_REF] Wazewski | Systèmes des Équations et des Inégalités Différentielles Ordinaires aux Seconds Membres Monotones et leurs Applications[END_REF]).

The following results may be easily proven.

Lemma 1. A functional g : (t, x, y) → g(t, x, y) is quasimonotone non-decreasing in x and non-decreasing in y iff it is mixed quasi-monotone non-decreasing in x, non-decreasing in y.

Lemma 2. If g is continuously differentiable with respect to

x and y, and ∀t ∈ R, ∀x ∈ R n , ∀y ∈ C n τ ∀i = j : ∂g i ∂x j ≥ 0, ∀(i, j) :

∂g i ∂y j ≥ 0, (4)
then g(t, x, y) is mixed quasi-monotone non-decreasing in x, non-decreasing in y.

The following theorem states a comparison principle for functional differential equations.

Theorem 1. Assume that: H1) ∀t ∈ R, ∀x ∈ R n , ∀y ∈ C n τ : f (t, x, y) ≤ g(t,
x, y), H2) g(t, x, y) is mixed quasi-monotone non-decreasing in x, non-decreasing in y, H3) g(t, x, y) is sufficiently smooth for (3) to possess, for every z t0 ∈ Ω ⊂ C n τ and for every t 0 ∈ R, a unique solution z(t) for all t ≥ t 0 .

Then: C1) For any x t0 ∈ Ω, the inequality x(t) ≤ z(t) holds for every t ≥ t 0 whenever it is satisfied for t ∈ [t 0τ, t 0 ]. In other words, ( 3) is a comparison system of (2) over Ω. C2) Moreover, if ∀t ≥ t 0 : 0 ≤ g(t, 0, ϕ 0 ) and z t0 ≥ 0, then 0 ≤ z(t).

Remark 2. One can refine the definitions given above by considering local comparison system and thus obtain a local version of this theorem (see [START_REF] Perruquetti | Sur la Stabilité et l'Estimation des Comportements Non Linéaires[END_REF], [START_REF] Perruquetti | Vector Lyapunov functions : recent developments for stability, robustness, practical stability and constrained control[END_REF]).

C. Linear cooperative systems with delays

Consider a linear system with constant delays and timevarying input

ẋ(t) = A 0 x(t) + N i=1 A i x(t -τ i ) + b(t), (5)
where x(t) ∈ R n is the state, x t ∈ C n τ for τ = max 1≤i≤N τ i where τ i ∈ R + are the delays; a piecewise continuous function b ∈ L n ∞ is the input; the constant matrices A i , i = 0, N have appropriate dimensions. The matrix A 0 is called Metzler if all its off-diagonal elements are nonnegative. The matrices

A i are called nonnegative if A i ≥ 0 (elementwise). The function g(t, x, x t ) = A 0 x(t)+ N i=1 A i x(t-τ i )+b(t) is mixed quasi- monotone non-decreasing in x, non-decreasing in x t if A 0 is Metzler and A i , i = 1, N are nonnegative.
Definition 3. The system ( 5) is called cooperative (or nonnegative [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF]) if A 0 is Metzler and A i , i = 1, N are nonnegative matrices.

The cooperative system (5) admits x(t) ∈ R n + for all t ≥ t 0 provided that x t0 ∈ C n τ + and b : R → R n + . Lemma 3. [START_REF] Dambrine | Stability Analysis of Time-Delay Systems[END_REF], [START_REF] Dambrine | Contribution à l'étude des systèmes à retards[END_REF], [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF] A cooperative system ( 5) is asymptotically stable for b(t) ≡ 0 for all τ ∈ R + iff there are p, q ∈ R n + (p > 0 and q > 0) such that

p T N i=0 A i + q T = 0.
Under conditions of the above lemma the system has bounded solutions for b ∈ L n ∞ with b(t) ∈ R n + for all t ∈ R. Lemma 4. [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF] Given the matrices A ∈ R n×n , R ∈ R n×n and C ∈ R p×n . If there is a matrix L ∈ R n×p such that the matrices A-LC and R have the same eigenvalues, then there is a P ∈ R n×n such that R = P (A -LC)P -1 provided that the pairs (A -LC, e 1 ) and (R, e 2 ) are observable for some e 1 ∈ R 1×n , e 2 ∈ R 1×n . This result was used in [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF] to design interval observers for LTI systems with a Metzler matrix R (in other words, the lemma establishes the conditions when the matrix A -LC is similar to a Metzler matrix). The main difficulty is to prove the existence of a real matrix P , and to provide a constructive approach of its calculation. In [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF] the matrix

P = O R O -1
A-LC , where O A-LC and O R are the observability matrices of the pairs (A -LC, e 1 ) and (R, e 2 ) respectively. Another (more strict) condition is that the Sylvester equation P A -RP = QC, Q = P L has a unique solution P provided that the pair (A, C) is observable (in this case there exists a matrix L such that λ(A) = λ(A -LC) = λ(R), that is equivalent to existence of a unique P ). Note that if the matrix A -LC has only real positive eigenvalues, then R can be chosen as diagonal or Jordan representation of A -LC.

D. Interval analysis

Lemma 5. Given a matrix A ∈ R m×n define A + = max{0, A}, A -= A + -A and |A| = A + + A -. Let x ∈ R n be a vector variable, x ≤ x ≤ x for some x, x ∈ R n , and A ∈ R m×n be a constant matrix, then

A + x -A -x ≤ Ax ≤ A + x -A -x. (6)
Lemma 6. Let A ≤ A ≤ A for some A, A, A ∈ R n×n and x ≤ x ≤ x for x, x, x ∈ R n , then

A + x + -A + x --A -x + + A -x -≤ Ax ≤ A + x + -A + x --A -x + + A -x -. (7)

III. MAIN RESULT

In this section, a design of an interval observer for the system (5) with varied input gains and constant delays is given, i.e. let [START_REF] Califano | On the observer canonical form for Nonlinear Time-Delay Systems[END_REF] be written in the form as follow:

ẋ(t) = A 0 x(t) + A 1 x(t -τ 0 ) + (B 0 ( 8
)
+δB 0 (t))u(t) + (B 1 + δB 1 (t))u(t -τ (t)), y = Cx + v(t),
where x(t) ∈ R n is the state, τ 0 is the constant delay and τ (t) is the time-varying uncertain delay; u(t) ∈ R m is the input; B 0 + δB 0 (t) is the input gains without delay, where δB 0 (t) is varied and uncertain, B 1 + δB 1 (t) is the input gains with time-delay (δB 1 (t) is varied and uncertain); y ∈ R p is the output available for measurements with a noise v ∈ L p ∞ . The matrices A 0 , A 1 , B 0 , B 1 , C and the matrix functions δB 0 (t), δB 1 (t) have appropriate dimensions.

Assumption 1. Let • x ∈ L n ∞ with x 0 ≤ x t0 ≤ x 0 for some x 0 , x 0 ∈ C n τ ; • ||v|| ≤ V for a given V > 0; • τ 0 ∈ R + are known, τ ≤ τ (t) ≤ τ ; • δB i (t) ≤ δB i (t) ≤ δB i (t) for all t ≥ t 0 for some known δB i , δB i ∈ L m ∞ i = 0, 1.
In this assumption it is supposed that the state of the system (5) is bounded with an unknown upper bound, but with a specified admissible set for initial conditions [x 0 , x 0 ]. The upper bound on the measurement noise amplitude V as well as the constant delays τ 0 are assumed to be given. All uncertainty of the system is collected in the gains δB 0 (t), δB 1 (t) and delay τ (t).

For the estimation purposes, the system (8) can be rewritten as follow:

ẋ(t) = (A 0 -L 0 C)x(t) + (A 1 -L 1 C)x(t -τ 0 ) + g(t), (9)
where

g(t) = L 0 y -L 0 V + L 1 y(t -τ 0 ) + L 1 V (t -τ 0 ) (10)
+ [B 0 + δB 0 (t)] u(t) + [B 1 + δB 1 (t)] u(t -τ (t)). Denote |L i | = L + i + L - i for i = 1, 0 and note that δB i (t)u(t) = m k=1 δB k i (t)u k (t), i = 1, 0,
where δB k i is the kth column of the matrix function δB i . Proposition 1. Let Assumption 1 be satisfied, then g(t) ≤ g(t) ≤ g(t)

V. CONCLUSION

An approach for interval estimation of systems with uncertain time-varying delays and uncertain input gains is presented. Numerical experiments performed for a thermic process confirm the efficiency of the proposed method. The classical observers without delay in general is not very precise when applied to systems with time-varying delay. Overcoming this obstacle, that is, determining interval observers that give a satisfactory estimation result when a small delay is present, is the main objective in this work. In addition, the appearance of two unknown parameters in the same time makes it more difficult to define the suitable interval estimation. In this work, the problem of unknown input gain of the system, which cannot be ignored when the practical problem is applied, is also considered.

Figure 1 .Figure 2 . 2 = δB 1 ≤ 2 .

 12212 Figure 1. Interval estimation of the output for the thermic model (the temperature)

where

Proof. According to Assumption 1 and the definition of

is completely known. To this end, it is necessary to calculate the interval bounds for δB 0 (t)u(t)+[B 1 + δB 1 (t)] u(t-τ (t)), using ( 6) and ( 7) we obtain:

The sum of these estimates provides the interval bounds g(t), g(t) such that g(t) ≤ g(t) ≤ g(t).

Following [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], the equation ( 9) and Proposition 1 let us propose the interval observer in the following form

Theorem 2. Let Assumption 1 be satisfied, the matrix A 0 -L 0 C be Metzler and the matrix

for all t > 0 provided that x(0) ≤ x(0) ≤ x(0). In addition, if there exist p, q ∈ R 2n + (p > 0 and q > 0) such that

Proof. Consider the estimation errors ε = x-x and ǫ = x-x, whose dynamics can be written as follows

Note that the initial conditions ǫ(0), ǫ(0) ∈ R n + and the dynamics of the errors are cooperative. By definition of g, g the signals d, d ∈ R n-p + . Thus ǫ(t), ǫ(t) ∈ R n + for all t > 0 provided that ǫ(0), ǫ(0) ∈ C n-p τ0 , the last relation is satisfied by the definition of x 0 and x 0 . To prove that the errors ǫ(t), ǫ(t) are bounded, as in [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF], consider for [START_REF] Mazenc | Exponentially stable interval observers for linear systems with delay[END_REF] the Lyapunov functional V : C n τ + → R + defined as

Let us stress that for any ϕ ∈ C n τ0+ the functional V is positive definite and radially unbounded, its derivative for ǫ takes the form (for ǫ the analysis is the same):

Thus for d = 0 the system is globally asymptotically stable, and since d ∈ L n-p ∞ (by construction and Assumption 1) one finds that the error ǫ is bounded (see [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF] or [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF] for the proof that in fact the system is input-to-state stable).

The performance of the proposed interval observer will be shown on example of thermic process.

IV. EXAMPLE

In this section, a model of thermic application is considered:

The system used is of the second order, where θ ∈ R is the temperature, u ∈ R is the input voltage, τ (t) > 0 is the timedelay, r > 0 is the time constant, k > 0 is the input gains, u 0 > 0 is a constant input value, y ∈ R is the output, which is the measured temperature.

The identification procedure, applied to the platform available at LAGIS laboratory, provided the interval estimates for the model parameters as below: To conclude, the dispersion of the parameter r values is rather narrow, that is why the median value r = 0.241 will be used next for modeling, while the deviations of k and τ cannot be neglected. Therefore, it is easy to verify that for x = [θ θ] this system can be written as [START_REF] Sename | Observer-based H∞ control for timedelay systems: A new LMI solution[END_REF] for