Hugo Delile
email: hugo.delile@cnrs.fr

André-Marie Dendievel

Anice Yari

Matthieu Masson

Cécile Miège

Brice Mourier

Marina Coquery

Legacy-micropollutant contamination levels in major river basins based on 2 findings from the Rhône Sediment Observatory 3

Keywords: Micropollutant yields, Long-term monitoring program, Large river basins, 12 Anthropogenic pressures, Persistent Organic Pollutants, Trace Metal Elements 13

For more than half a century, chemical contamination has progressively spread to all the world's 15 large river basins. Large river outlets integrate multiple anthropogenic pressures in watersheds, 16 making them the largest source of sediment-bound contaminants to continental shelf areas. 17 However, comparing particulate micropollutant contaminations between the large river basins 18 is a challenging task, especially due to the scarcity of long-term river monitoring programs.

19

Here we address this issue, with a focus on legacy particulate micropollutants 20 (polychlorobiphenyls [PCBi], polycyclic aromatic hydrocarbons [PAHs], and trace metal 21 elements [TME]) yields. For this purpose, we employed a bottom-up multiscale approach to 22 chemical contamination in river basins that takes micropollutant yields measured in the Rhône 23 River sub-basins (France) as a benchmark of other large river basins. Data on the Rhône River 24 basin came from a unique 10-year-long monitoring program within the Rhône Sediment 25 Observatory (OSR), and were compared to data gathered on 18 major worldwide river outlets.

INTRODUCTION 33

For more than five decades, freshwaters worldwide have experienced large-scale chemical 34 deterioration, especially by legacy pollutants, to the extent that water quality concerns have 35 become a major challenge for humanity in the twenty-first century (Meybeck and Helmer, 1989; 36 [START_REF] Schwarzenbach | Global Water 936 Pollution and Human Health[END_REF]UNEP-DHI and UNEP, 2016;Damania et al., 2019). As a result, The dataset used to assess inter-decadal trends includes several types of sediment matrices 205 (Tab. 2) that are likely to affect micropollutant concentrations (effect of grain-size distribution 206 and/or organic matter concentrations). For this reason, in figure S1, we compared 207 micropollutant concentrations between suspended (SPM) and deposited sediments (including 208 bed and flood deposits, and sediment cores) for the 2010-2019 period. Overall, organic 209 micropollutant concentrations were 25% lower in SPM than in deposited sediments, whereas 210 TME were 30% to 100% higher in SPM. Even though these differences are not negligible, 211 concentrations remain similar in magnitude and were used for the assessment of the inter-212 decadal trends. However, the dataset used for the assessment of intra-decadal trends was 213 exclusively from SPM sediments (Tab. 2), since at this fine temporal scale, mixing various 214 sediment matrices would undermine the results.

(Arles/Beaucaire station, i.e. 120 kilometers from the Rhône River outlet) for TME (Cd,Cr,217 Cu, Hg, Ni, Pb and Zn), 7 indicator PCBi (congeners 28, 52, 101, 118, 138, 153, and To assess micropollutant yields, we estimated the annual sediment fluxes discharged at 303 the outlet of the selected rivers. Accurate estimation of sediment fluxes remains a challenging 304 issue (Mouyen et al., 2018;Poulier et al., 2019;Hackney et al. 2020), especially at global scale, 305 because less than 10% of the river mouths worldwide have monitoring of sediment delivery 306 (Syvitski et al., 2005;Bravard, 2018). In addition, in-situ sediment discharge measurements are 307 highly uncertain (spatially sparse and temporally discontinuous data, different-measurement 308 time frequencies, heterogeneous measuring techniques) (Mouyen et al., 2018). We therefore 309 used the modeled annual fluvial sediment fluxes (Nienhuis et al., 2020) for all of the large rivers 310 included in this study (Tab. 1). These data were produced using WBMSed (Cohen et al., 2013, 311 2014) and BQART (Kettner and Syvitski, 2008) methods. This approach using modelled 312 estimates of sediment flux at the large river outlets appears particularly suitable for our study, 313 since it catches the multiyear average sediment fluxes as well as inter-annual and intra-annual 314 trends (Cohen et al., 2013).

315

Then, annual micropollutant fluxes were calculated by multiplying the sediment fluxes 316 by the median concentration of each micropollutant (Fig. S2) for each river on the basis of 317 sediment samples collected at the outlet of the rivers. Based on our homemade micropollutant 318 concentrations database, Table S2 shows specific concentrations (median and range) of 319 micropollutants (mg or μg kg -1 dry weight), inter-annual particulate micropollutant fluxes (kg 320 or t yr -1 ), yields (g or kg km -² yr -1 ) and per capita data (mg or g inhab magnitude. Thus, the spatial distribution of the Ʃ 12-16 PAH yields, for instance, was divided in 343 three order-of-magnitude classes, i.e. (1) 1-10, (2) 10-100 and (3) >100 g km -² yr -1 .

344

RESULTS

345

We present in this section results on the temporal and spatial trends of contamination in the

346

Rhône River basin and on the assessment of the relationship between the micropollutants 347 yields and SPM yields of large river basins.

368

The relative weights of C micropol and F SPM in F micropol variations from one decade to the Zn, Cd and Cu (Fig. 5). In addition, the first TME group is significantly correlated with SPM agriculture [START_REF] Schwarzenbach | Global Water 936 Pollution and Human Health[END_REF]Landrigan et al., 2018). This growing demand for 522 energy and natural resources is driven by uncontrolled growth of cities and large-scale 523 urbanization in these countries [START_REF] Wilkinson | Energy, energy efficiency, 1008 and the built environment[END_REF]UNESCO, 2009;Shen et al., 2013).

524

The environmental quality improvement of the Rhône River basin during the 2010s is 525 apparent as a leftward shift in Figure 8 oil combustion, etc.), the calculated Hg emission yield of 11 g km -² yr -1 is close to our appraisal 586 of 12 g km -² yr -1 (Table S2) and to other estimates, such as 9.3 g km -² yr

611

Legacy contaminated sediments along the river continuum pertain to the issue of (i)

CONCLUSIONS 617

We compared contamination levels in the large river basins using particulate micropollutant 618 yields, which were explored through a bottom-up multiscale river basins approach. In this "Forests" for forest areas, "Bare" for bare areas, "Vegetal" for shrub and/or grassy vegetation,

1054

"ICA" for industrial and commercial areas, and "Urban" for urban areas. On the whole, the Agri" for agricultural lands, "Forests" for forest areas, "Bare" for bare areas, "Vegetal" for shrub and/or grassy vegetation, "ICA" for industrial and commercial areas, and "Urban" for urban areas. On the whole, the dominant factor F1 results from the contribution of all micropollutant (except for ∑5PCBi) and SPM yields. Factor F2 opposes the anthropogenic-related land use categories (urban, agricultural, industrial and commercial areas) against the forests land-use category. -66 -75 -76 -83 -36 -53 -37 -53 -26 -45 -21 -41 -42 -57 -26 1990s-2000s -43 -59 -31 -50 -54 -67 +63 +18 -35 -53 -21 -43 -6 -32 -7 -33

37-

  over the last ten years, this chemical contamination issue has become part of a broader concept The intra-decadal trends refer to the 2008-2019 period based on the dataset gathered 125 monthly or every two weeks by the OSR monitoring program. Trends are therefore 2008-2010, TME, PAHs and PCBi data came from research labs and monitoring 200 agencies studying deposited sediments, i.e. bed and flood deposits, and sediment cores; 201 -after 2008-2010, TME, PAHs and PCBi data came primarily from SPM measured under 202 the OSR monitoring program. A small amount of data acquired on deposited sediments 203 was also used to build a composite record for the 2010-2019 period.

  204

Figure 2

 2 Figure2shows the pluri-decadal trends in particulate micropollutant concentrations and

  to 2010s, ΔC ∑7PCB -39%, and ΔC ∑12PAH -38% on average) and fluxes 360 (1990s to 2010s, ΔF ∑7PCB -37%, and ΔF ∑12PAH -35% on average) (Fig.2A-D and Tab. 3).361The time-course of PCB and PAH fluxes over the period reproduces the same pattern as the 362 concentrations. In contrast to the POPs, TME fluxes showed a similar pattern of change to the 363 SPM flux trend. TME and SPM fluxes decreased significantly with similar rank order of 364 magnitude between the 1990s-2000s (ΔF TME -34% and ΔF SPM -28%), whereas ΔC TME 365 decreased slightly ( -8%) over the same period. Thereafter, TME and SPM fluxes increased 366 sharply between 2000 and 2010s (ΔF TME +21% and ΔF SPM +44%) while ΔC TME 367 experienced a slight decline ( -8%) (Fig. 2E-F and Tab. 3).

  431 fluxes (R values 0.93-0.98; p-value < 0.05), which is not the case for the second group (R 432 values 0.48-0.68; p-value > 0.05). The location of the river stations into the PCA differentiates 433 the northern and southern areas of the Rhône River basin: the Upper Rhône and Saône rivers in 434 the north showing the lowest micropollutant yields, and the Isère and Durance rivers in the 435 south characterized by natural land use and SPM flux-bound TME yields. The Gier River 436appears to be the most contaminated basin, especially for organic micropollutants yields (Fig.4374), and also the most impacted by anthropogenic land use (Figs.1 and 5).

  findings from the Rhône River basin on the main patterns of micropollutant 441 yields, we next extrapolated that knowledge of SPM fluxes to assess micropollutant yield signal 442 of the large river basins. The maps of micropollutant yields show that some TME yields follow 443 spatial trends of SPM yields (Fig.S3), whereas PAH and PCB yields seem unconstrained by 444 SPM yields (Fig.6). River basins in high-income countries (i.e. Europe and North America) SPM yields (< 66 t km² yr -1 ) (except for the Rhine River basin), while river 446 basins in low-income countries (i.e. Asia and South America) show the highest SPM yields (> 447 100 t km² yr -1 ) (except for the Niger River basin)(Figs. 6 and S3, and Tab. 1).448The potential control of SPM fluxes on micropollutant yields was assessed through the 449 correlation circle of the PCA (Fig.7A, see details on the PCA method in the Supplementary 450 Text) and the correlation matrices between micropollutant and SPM yields (Tab. S3), and 451 between micropollutant yields and median concentrations (Tab. S4). We identified three groups 452 of micropollutant yields: 453 1) Anthropogenic activity-driven PCB, PAH, Cd and Hg yields: uncorrelated with SPM 454 yields (R 0.15-0.40, p-value > 0.05), and positively correlated to their median 455 concentrations (R 0.6-0.8, p-value < 0.05); 456 2) SPM flux-driven Co, Cr, Ni and Cu yields: positively correlated to SPM yields (R 457 0.8-1, p-value < 0.05), and uncorrelated with their median concentrations (R 0.2activity/SPM flux-driven As, Zn and Pb yields: positively correlated to 460 SPM yields (R 0.5-0.8, p-value < 0.05) and to their median concentrations (R 0the Rhône River's intra-basin contamination has shown a striking spatial 466 heterogeneity between the northern and southern areas of the Rhône River basin. For instance, 467 except for Zn, the first group of TME concentrations (As, Co, Cr, Ni and Zn) of the Rhône 468 River basin matches the specific chemical fingerprint of the Isère River's SPM (Delile et al., 469 2020). We can therefore assume a process of enhanced soil erosion in specific Isère River sub-, Co and Cr element-rich geological basements, leading to increasing 471 concentrations of these TME over the past decade (2010s). However, only additional TME 472 geochemical tracing methods, such as Pb, Cu and Zn isotopic fingerprinting techniques (Bird, 473 2011) or geochemical fingerprinting (Bégorre et al., 2021; Dabrin et al., 2021), could capture 474 specific sources of TME in sediments. This heterogeneous pattern of the micropollutant yields 475 of the Rhône River basin suggests that calculated micropollutants yields of the large river basins 476 need to be considered with caution, in terms of the sources of the chemical fluxes. Such fluxes 477 should not be considered as uniformly deposited over the entire surface of the river basins. Our 478 reconstructions of the temporal trends of particulate micropollutant concentrations and fluxes 479 at the Rhône River outlet since the 1980s show that micropollutant concentrations have to vary 480 by more than 30% to affect micropollutant fluxes. As a result, this threshold defines the switch 481 from SPM flux-driven to anthropogenic pressure-driven micropollutant yields. Applied to the 482 other 18 large river basins worldwide, this balance between micropollutant concentrations and 483 SPM fluxes suggests a progressive anthropogenic origin of the micropollutant yields as follows: 484 (i) SPM flux-driven Co, Cr, Ni and Cu yields, (ii) anthropogenic activity and SPM flux-driven 485 As, Zn and Pb yields, and (iii) anthropogenic activity-driven PCB, PAH, Cd and Hg yields.

  of economic development on the global distribution of micropollutants yields 506 Considering all micropollutant yields, the distribution of the large river basins within the PCA 507 defines three statistically-based mixing lines showing a gradient of overall contamination level 508 from the least contaminated river basins in the bottom-left corner of the graph up to the most 509 contaminated river basins in the upper-right part of the graph (Fig. 7B). The slope mixing lines 510 depend of the relative weight between the anthropogenic origin of the micropollutant yields 511 (Group 1-based mixing line) and the contribution of SPM fluxes to micropollutant yields 512 (Groups 1 + 2 +3-based mixing line). The mixing lines split river basins into two groups: the 513 most contaminated rivers located in developing countries (Asia and South America), and the 514 least contaminated rivers located in developed countries (Europe and North America).515Based on the long-term trends of the Rhône River basin contamination, it appears that the 516 overall chemical contamination level during the 1980s was similar to that currently observed in 517 countries undergoing economic transition. According to previous works, the highest 518 contamination levels, tied to the developing low-to-middle-income countries, result from both energy required for petroleum-powered vehicles and electricity 520 generation and natural resources (e.g., mining) or else from deforestation due to intensive 521

612

  contaminant residence times along river corridors(Pizzuto, 2014; Sutfin and Wohl, 2019; 613[START_REF] Wohl | Legacy effects on sediments in river corridors[END_REF][START_REF] Wohl | Forgotten Legacies: Understanding and Mitigating Historical Human 1012 Alterations of River Corridors[END_REF]), (ii) total time for contaminant transfer to river outlets, and (iii) the 614 theoretical decontamination time of a river basin(Singer et al., 2013; Gateuille et al., 2014,

  616

  compared micropollutant yields based on a unique dataset from a 10-year-long-620 monitoring program on the Rhône River sub-basins, then extrapolated up this approach to other 621 major river basins in order to assess the spatial representativeness of this chemical 622 contamination proxy. 623 Our reconstructions of the temporal trends of particulate micropollutant concentrations 624 and fluxes at the Rhône River outlet since the 1980s show that micropollutant concentrations 625 have to vary by more than 30% over 10 years to affect micropollutant fluxes. As a result, this 626 threshold defines the tipping point marking the switch from SPM flux-driven to anthropogenic 627 pressure-driven micropollutant yields. Applied to the other 18 large river basins worldwide, this 628 balance between micropollutant concentrations and SPM fluxes suggests a progressive 629 anthropogenic origin of the micropollutant yields as follows: (i) SPM flux-driven Co, Cr, Ni 630 and Cu yields, (ii) anthropogenic activity and SPM flux-driven As, Zn and Pb yields, and (iii) 631 anthropogenic activity-driven PCB, PAH, Cd and Hg yields. The distribution of the magnitude 632 of chemical contamination in the large river basins is related to the economic development stage 633 of the local country. Our results show an overall contamination gradient splitting the most 634 contaminated river basins in developing and industrializing low-income and middle-income 635 countries (Asia and South America) from the least contaminated rivers in developed high-636 income countries (Europe and North America). This kind of spatial pattern in surface waters-637 related chemical contamination fits well with the Environmental Kuznets Curve theory. Indeed, 638 our results show how a prevailing model of economic development implies rampant 639 urbanization supported by both increasingly high energy and natural resource consumption. In 640 turn, these conditions constitute major sources of chemical contamination in the world's large 641 river basins. In this respect, the long-term trend of contamination in the Rhône River basin 642 shows an overall chemical contamination level in the 1980s that was similar to levels observed 643 today in countries undergoing economic transition (Asia and South America).

Figure 1 . 1030 Figure 2 .Figure 3 .Figure 4 .Figure 5 .

 110302345 Figure 1. (A) Map of the Rhône River basin with its main tributaries and monitoring stations

Figure 6 .Figure 7 . 1084 Figure 8 .

 6710848 Figure 6. Maps of the ∑ 12-16 PAH, ∑ 7 PCBi, Hg, Cd and SPM yields, and GDP per capita in the

  Long-term decadal variations (in %) in inter-annual particulate micropollutant 1094 concentrations (ΔC) and fluxes (ΔF) and SPM fluxes at the Rhône River outlet. Orange cells 1095 show the micropollutant flux variations that are mainly dependent on micropollutant 1096 concentration variations, while grey cells highlight micropollutant flux variations that are 1097 mainly dependent on SPM flux variations. The bottom part of the table shows the correlation 1098 coefficients for decadal micropollutant fluxes and concentrations (first line) and for 1099 micropollutant fluxes and SPM fluxes (second line).

Figure 2 .Figure 3 .Figure 4 .Figure 5 .

 2345 Figure 1. (A) Map of the Rhône River basin with its main tributaries and monitoring stations for water discharge, suspended particulate matter, and associated micropollutants. (B) Land-use distribution (in %) in the basin of the Rhône River (Arles/Beaucaire station) and its main tributaries. 299x288mm (400 x 400 DPI)

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Maps of the ∑12-16PAH, ∑7PCBi, Hg, Cd and SPM yields, and GDP per capita in the major world river basins (1-Rhône, 2-Garonne, 3-Loire, 4-Seine, 5-Volga, 6-Danube, 7-Elbe, 8-Rhine, 9-Pearl (Zhujiang), 10-Liao, 11-Yangtze (Changjiang), 12-Yellow (Huang He), 13-Mekong, 14-Ganges-Brahmaputra, 15-Mississippi, 16-Colorado, 17-St. Lawrence, 18-Amazon, 19-Niger). 424x425mm (350 x 350 DPI)

F

Assessment of the inter-annual micropollutants yields of the world's large river 301 basins 302

  weighted by the corresponding watershed size to get the micropollutant yields (g or kg km -² yr -1 299 for POPs or TME, respectively) at the Rhône River basin and sub-basin scales.

	298
			F o F o F o r r r
			P P P e e r e e r e e r
			R R R e v i e v i e v i e e e w w w
					180; noted
	218	Ʃ 7 PCB, and 12 priority PAHs (anthracene, benz[a]anthracene, benzo[a]pyrene,
		benzo[b]fluoranthene,	benzo[ghi]perylene,	benzo[k]fluoranthene,	chrysene,
		dibenz[a,h]anthracene, fluoranthene, indeno[1,2,3-cd]pyrene, phenanthrene, pyrene; denoted
	221	with the abbreviation Ʃ 12 PAH). All of these data were produced by monitoring authorities (the

222

Rhône-Mediterranean-Corsica water agency) and the OSR monitoring program. Our database 300 4.3.

Table 3 presents

 3 352 specific variations (in percent) of micropollutant concentrations and fluxes and SPM fluxes 353 from one decade to the next, noted ΔC micropol , ΔF micropol and ΔF SPM , respectively. The general

	354	time-course of micropollutant concentrations and fluxes follows a similar decreasing trend
	355	between 1980s-1990s: -58% for ΔC ∑7PCB and -69% for ΔF ∑7PCB , and -42% for mean ΔC TME and

356

-57% for mean ΔF TME .

357

After the 1990s, the organic and inorganic micropollutant concentrations and fluxes 358 follow distinct patterns. POPs continued to decline in similar and stable proportions, both in

1.2. Short-term trends in micropollutant contamination over the past decade

	372	weight of ΔC micropol and ΔF SPM in ΔF micropol : (1) concentration-driven micropollutant fluxes for
	373	∑ 7 PCB, ∑ 12 PAH, Cd, Cr and Hg (orange cells, and statistically significant correlation of the time-series (2008-2016) decreased at a mean annual rate of 1.4 μg kg -1 (i.e. -9% per year;
	374 398	coefficients of F micropol vs. C micropol in Tab. 3), (2) SPM flux-driven micropollutant fluxes for Cu Fig. 3). This trend is in line with the general pattern of decreasing PCB contamination of the
	375 399	and Ni (grey cells, and statistically significant correlation coefficients of F micropol vs. F SPM in Rhône River waters that began in the 1990s (Desmet et al., 2012; Mourier et al., 2014; Babut
	376 400	Tab. 3), and (3) alternately concentration-driven and SPM flux-driven micropollutant fluxes for et al., 2016; Liber et al., 2019; Dendievel et al., 2020a) (Fig. 2A). Nonetheless, the trend showed
	377 401	Pb and Zn (orange and grey cells, and statistically significant correlation coefficients of F micropol a change-point in October 2013, with a slight increase in PCB concentrations of about +1.6 μg
	378 402	vs. C micropol and of F micropol vs. F SPM in Tab. 3). According to Table 3, SPM flux-driven kg -1 per year (i.e. +9%) (Fig. 3).
	379 403	micropollutant fluxes switch to concentration-driven micropollutant fluxes when ΔC micropol is 5.2
	380 381	F o r higher than ± 30%. In other words, variations of micropollutant concentrations have to vary F o by more than 30% in order to affect micropollutant fluxes. Below this threshold, SPM fluxes r
	382	P primarily control micropollutants fluxes. P
	384	e e r Figure 3 shows the temporal evolution of particulate micropollutant concentrations at the Arles e e r
	385 386	R e v i R station (Rhône River outlet, Fig. 1) over the past decade (2010s). Mean monthly water discharges and associated temporal trends over the total length of the time-series are also shown e v i
	387	e in order to capture the relationship between the water discharge trends and the micropollutant e
	388	w concentrations trends. Overall, the results highlight three micropollutant trend groups: (1) w
	389	increasing concentrations for As, Co, Cr, Ni and Zn, (2) decreasing concentrations for Ʃ 7 PCB,
	390	and (3) no trend for Ʃ 12 PAH, Cd, Cu, Hg and Pb (Fig. 3). Given that the mean monthly water
	391	discharge does not show statistically significant temporal trends, it cannot lead to an increase
	392	or decrease in micropollutant concentration trends.
	369 393	next were assessed (see Tab. 3) by comparing (i) the decadal variations (in %) of inter-annual The comparison between the increase in annual rates of TME concentrations (red values
	370 394	particulate micropollutant concentrations (ΔC) and fluxes (ΔF), and SPM flux and (ii) their in Fig. 3) of the first group (As, Co, Cr, Ni and Zn) and their respective median value (Tab. S2)
	371 395	respective correlation coefficients. We defined three micropollutant groups by assessing the indicates an average annual rate of increase of 7% per year.

383

5.

. Spatial trend in Rhône River basin contamination

	420	the contribution of mean TME yields to the F1 component (7.5%) is twice as large as the
	404	Figure 4 is a map of the particulate micropollutants yields of the whole Rhône River basin F o
	405	and its main sub-basins. It shows a sharp contrast between a less contaminated area in the North r
	406 407 408 430	River, Gier River and Durance River basins). This contrast appears more clearly for TME yields, which have a spatial distribution pattern that matches the features of intra-basin SPM r e e (Saône River and Upper Rhône River basins) and a more contaminated area in the South (Isère P
	409 410	R yields (Fig. 4). As a result, TME yields in the different tributaries seem to be sensitive to variation in SPM yields. The SPM yields in the Rhône River basin come mainly from the Isère e v i
	411	and Durance river basins, since they are known to be the largest suppliers of the inter-annual e
	412	SPM flux discharged into the Mediterranean Sea from the Rhône River outlet. Each of these w
	413	two tributaries accounts for 31% of the total inter-annual SPM flux of the Rhône River, while
	414	the Upper Rhône River, Saône River and Gier River account for 11%, 6%, and 0.1%,
	415	respectively (Delile et al., 2020).
	416	Figure 5 is a Principal Component Analysis (see details on the PCA method in the
	417	Supplementary Text) combining the micropollutants yields, population density and land use of
	418	each station (Fig. 1B). The first principal component (F1 ~ 56%) results from the contribution
	419	of all micropollutants yields, meaning that F1 tracks the overall contamination level. However,

421 contribution of organic micropollutants (2.7%). Organic micropollutants are more involved in 422 the F2 positive values, since PAH and PCB yields contribute on average about 8% to the F2 423 component (only 2.5% for mean TME contribution). The F2 axis shows a striking contrast 424 between anthropogenic land use (positive values) as agricultural lands, urban areas and 425 industrial and commercial areas (ICA) on one side and natural land use (forests, bare, and 426 shrub/vegetation) (negative values) on the other (Fig. 5). Population density is highly correlated 427 to urban areas and ICA (R > 0.93 and 0.89, respectively; p-value < 0.05). Based on the F2 428 component, TME yields divide into two groups: first, a natural land use-related TME group 429 with Cr, Co, Ni and Hg, and a second anthropogenic land use-related TME group with As, Pb,

  We argue that the diversity of the geological basements of the Rhône River sub-Nevertheless, the clustering of TME demonstrates that the proportion of natural and 500 anthropogenic inputs is not the same for all TME. Indeed, at the worldwide river basins scale, 501 the anthropogenic source is the dominant contributor to Hg and Cd whereas the natural source 502 is the dominant contributor to Ni, Co, Cr and Cu. The Rhône River monitoring network data

	495	contamination. Indeed, as it stands, TME yields reflect a mixture comprising both the TME
	496	fraction coming from the local geological environment and TME fraction from the
	497	anthropogenic activity. Therefore, TME yields would be more suitable to track anthropogenic
	498	contamination of river basins if local geological effects on TME yields could be removed.
		F o
		r
		P
		e e r
		R
		e v i
		e
		w
	491	basins captured in Rhône Sediment Observatory data is the decisive factor for explaining this
	492	distinction between Hg and Cu yields.

486

It is striking to note that these large river basins-bound micropollutant yield groups are 487 close to those identified in the Rhône River sub-basins since PCB, PAH, As, Pb, Zn, Cd and 488 Cu yields are related to their contamination level, while Cr, Co, Ni and Hg yields are more 489 related to their SPM yields. Only Hg and Cu yields do not belong to the same group. This is a 490 key point. 493 As a result, regarding TME, the lack of geochemical background data for each river basin 494 is a significant barrier to considering TME yields as a consistent proxy for assessing river basin 499 503

shows the tipping-point switch from a predominantly natural to anthropogenic TME yields 504 requires TME concentrations that are 30% higher than the local geochemical backgrounds.

Comparison of the contamination levels in the world's large river basins

  towards the cluster of less-contaminated rivers. This 526 substantial decline since the 1980s is part of a global decline in POPs(Breivik et al., 2002a,b or GDP rises. However, this theory has been criticized due to, for example, the 545 unsuitable statistical analyses of time-series(Stern et al., 1996; Stern, 2004) or the 546 environmental and economic activity indicators on which it is based(Dinda, 2004). The heavy industrial legacy of the Rhine and Elbe river basins(Heise et al., 570 2004; Förstner et al., 2004) could explain the high PAH contamination of these areas.The spatial distribution pattern of TME yields is broadly similar to that of PAH, with the 572 highest levels found in Asia, followed by river basins in Europe and North America. However,

	544 income and/547 The observed relationship between country income level and SPM yield derives from 6.3. 548 569 of Asian countries. 573 the Amazon River basin shows an equal if not higher level of TME contamination than river
	574	basins in Europe and North America (Figs. 6 and 8). Once again, this pattern is consistent with
	549	well-known anthropogenic stresses such as agriculture-induced deforestation and river
	575	worldwide TME emissions estimates, since Asia accounts for between 40% and 60% of the
	550	damming (Syvitski and Kettner, 2011; Bravard, 2018; Nienhuis et al., 2020).
	576	total depending on the TME, followed by Europe, North America and South America which
	577	F o o F contribute between 10% and 20% depending on the TME (Pacyna and Pacyna, 2001; Streets et
		r r
		P P
		e e r e e r
		R R
		e v i e v i
		e e
		w w
	562	Europe and North America show lower PAH yields (Figs. 6 and 8) thanks to the implementation
	563	of emissions reduction technologies (Shen et al., 2011, 2013) and a series of legislative policies
	539 564	successive economic stages of development. Such a trend fits rather well with the theoretical in force since the 1970s, first with the 1979 Convention on Long-Range Transboundary Air
	540 565	framework of the Environmental Kuznets Curve (EKC) (Grossman and Krueger, 1991, 1995; Pollution (CLRTAP) (UNECE, 1979) and then the Aarhus Protocol on POPs (United Nations,
	541 566	Shafik and Bandyopadhyay, 1992). According to EKC theory, this relationship, which is 1998). The implementation of this regulatory framework has brought about a sharp decrease in

527

and 2007 for PCB;

Shen et al., 2011Shen et al., , 2013

for PAH) and TME releases

(McConnell et al., 2019;

528

Legrand et al., 2020)

. This decline may result from national (see French regulation timeline for 529 PCBi in Fig.

2

) and international implementations of environmental regulations (e.g. PCBi 530 listed in the Stockholm Convention [UNEP, 2001] and PAHs in the Aarhus Protocol on 531 Persistent Organic Pollutants [United Nations, 1998]) in many developed countries (Miller and 532 Orbock Miller, 2007). Indeed, other studies have shown that the lower contamination level of 533 the river basins located in developed countries results from (i) efforts to mitigate industrial, 534 municipal and household wastewaters, (ii) the commissioning of sanitation and water supply 535 infrastructures, and (iii) national and international environmental regulations on emissions 536 control and the use of chemical products (UNESCO, 2009; Schwarzenbach et al., 2010; UNEP-537 DHI and UNEP

[START_REF] Schwientek | Integrated monitoring 942 of particle associated transport of PAHs in contrasting catchments[END_REF] Landrigan et al., 2018; Damania et al., 2019)

.

538

This general model seems to show an environmental quality improvement depending on 542 assumed to take the form of an inverted U-shape curve, comes from a period of environmentally 543 damaging economic growth followed by improvement in environmental quality as per capita 551 Polycyclic aromatic hydrocarbons and TME yields mapping well depict the relevance of 552 consumption of natural resources and energy for the economic development of our modern 553 societies (Figs. 6 and S3). In both cases, the highest micropollutant yields of Asian countries 554 (Figs. 6 and 8) are consistent with their atmospheric micropollutant emissions, since they 555 contribute around 50% of the global ∑ 16 PAH (Shen et al., 2013) and TME emissions (Pacyna 556 and Pacyna, 2001). It has been estimated that almost two thirds of the global ∑ 16 PAH emissions 557 come from residential/commercial burning of biomass, and this source accounts for about 80% 558 of emissions from South-East Asia (Shen et al., 2013). As underlined above, recent surging 559 urbanization is considered as the main driver of this high PAH contamination in developing 560

countries, because it results from extensive use of biomass for cooking and heating but also 561 from petroleum consumption by road vehicles

(Shen et al., 2011(Shen et al., , 2013))

. River basins in Western 567 ∑ 12 PAH yields in the Rhône River basin, from 100.5 g km -² yr -1 in the 1990s to 34.6 g km -² yr -1 568 in the 2010s. However, the Rhine and Elbe river basins show similar PAH yield levels to those 571 578 al., 2017; Kubier et al., 2019). The two main sources of Hg, Cd, As, Zn, Cu, and Ni 579 contamination are non-ferrous metal production and non-vehicle fossil fuel combustion 580 (Pacyna and Pacyna, 2001; Streets et al., 2017). For example, during the 1990s, gold mining in 581 the Brazilian Amazon released around 130 t of Hg into the environment every year, of which 582 about half of that amount reached surface waters (Pfeiffer et al., 1993; Lacerda and Salomons, 583 1998). By adding the other minor anthropogenic Hg sources (e.g. chemicals manufacturing, 584 electrical and measuring equipment, Cu, Zn and Pb smelting, silver (Ag) production, coal and 585

  Pardé M. 1925. Le régime du Rhône. Revue de Géographie Alpine 13 (3): 459-547 (in French). Riverine transport of water, sediments and pollutants to the Mediterranean Sea.

	844 894 994	244 119 Environment Program Chemicals, Geneva, Switzerland.
	845 895	Mai B, Zeng EY, Luo X, Yang Q, Zhang G, Li X, Sheng G, Fu J. 2005. Abundances,
	846 896	Invisible Water Crisis. Washington, DC: World Bank. depositional fluxes, and homologue patterns of polychlorinated biphenyls in dated Pettitt A.N. (1979). A non-parametric approach to the change-point problem. Applied Statistics
	897	28, 126-135.
	898	Pfeiffer WC, Lacerda LD, Salomons W, Malm O. 1993. Environmental fate of mercury from
	899	gold mining in the Brazilian Amazon. Environmental Reviews: 26-37
	900	Pizzuto J.E. 2014. Long-term storage and transport length scale of fine sediment: Analysis of
	901	a mercury release into a river. Geophysical Research Letters 41 (16): 5875-5882
		F o F o F o F o F o F o F o F o F o F o F o F o F o F o F o
		r r r r r r r r r r r r r r r
		P P P P P P P P P P P P P P P
		e e r e e r e e r e e r e e r e e r e e r e e r e e r e e r e e r e e r e e r e e r e e r
		R R R R R R R R R R R R R R R
		e v i e v i e v i e v i e v i e v i e v i e v i e v i e v i e v i e v i e v i e v i e v i
		e e e e e e e e e e e e e e e
		w w w w w w w w w w w w w w w
	812 912	Guha N, Baan R, Mattock H, Straif K, et al. 2013. Carcinogenicity of polychlorinated
	813	biphenyls and polybrominated biphenyls. The Lancet. Oncology 14 (4): 287-288
	814	Le Bescond C, Thollet F, Poulier G, Gairoard S, Lepage H, Branger F, Jamet L, Raidelet N,
	815	industrial palaeochannel sediment cores along the Rhône River (France). Data in Brief Radakovitch O, Dabrin A, et al. 2018. Des flux d'eau aux flux de matières en suspension Pacyna JM, Pacyna EG. 2001. An assessment of global and regional emissions of trace metals
	741 816 891	32: 106256 et de contaminants associés : gestion d'un réseau de stations hydro-sédimentaires sur le to the atmosphere from anthropogenic sources worldwide. Environmental Reviews:
	892	269-298
		Then, starting in

742 Desforges J-P, Hall A, McConnell B, Rosing-Asvid A, Barber JL, Brownlow A, De Guise S, 743 Eulaers I, Jepson PD, Letcher RJ, et al. 2018. Predicting global killer whale population Landrigan PJ, Fuller R, Acosta NJR, Adeyi O, Arnold R, Basu N (Nil), Baldé AB, Bertollini 809 R, Bose-O'Reilly S, Boufford JI, et al. 2018. The Lancet Commission on pollution and 810 health. The Lancet 391 (10119): 462-512 811 Lauby-Secretan B, Loomis D, Grosse Y, El Ghissassi F, Bouvard V, Benbrahim-Tallaa L, 817 Rhône. La Houille Blanche (3): 63-70 (in French). 818 Legrand M, McConnell JR, Lestel L, Preunkert S, Arienzo M, Chellman NJ, Stohl A, Eckhardt 893 Panayotou T. 2000. Economic Growth and the Environment. CID Working Paper Series 56: 902 Poulier G, Launay M, Le Bescond C, Thollet F, Coquery M, Le Coz J. 2019. Combining flux 903 monitoring and data reconstruction to establish annual budgets of suspended particulate 904 matter, mercury and PCB in the Rhône River from Lake Geneva to the Mediterranean 905 Sea. The Science of the Total Environment 658: 457-473 906 Qi W, Müller B, Pernet-Coudrier B, Singer H, Liu H, Qu J, Berg M. 2014. Organic 907 micropollutants in the Yangtze River: Seasonal occurrence and annual loads. Science of 908 The Total Environment 472: 789-799 909 Rasmussen JJ, Wiberg-Larsen P, Baattrup-Pedersen A, Monberg RJ, Kronvang B. 2012. 910 Impacts of pesticides and natural stressors on leaf litter decomposition in agricultural 911 streams. The Science of the Total Environment 416: 148-155 995 UNEP/MAP -United Nations Environment Programme -Mediterranean Action Plan. 996 2003. 997 Technical Reports 141. Athens.

Table 1 .

 1 Main physical and human characteristics of the large river basins investigated in this

	1088
	1089	study. Data sources are shown in footnotes.

1090

Table 2 .

 2 River sediment matrices and data sources used to assess the temporal trends in

	1091	particulate micropollutant concentrations and fluxes at the Rhône River outlet listed by
	1092	micropollutant family and the decades considered.

Micropollutant concentrations data were gathered from the lower part of the Rhône River

http://mc.manuscriptcentral.com/hyp Hydrological Processes

Damania R, Desbureaux S, Rodella A-S, Russ J, Zaveri E. 2019. Quality Unknown : The

Desmet M, Mourier B, Mahler BJ, Van Metre PC, Roux G, Persat H, Lefèvre I, Peretti A,

Gioia R, Dachs J, Nizzetto L, Berrojalbiz N, Galba C, Del Vento S, Méjanelle L, Jones KC.

Supplementary Text. All the data gathered on our homemade micropollutant concentrations Darken P.F. (1999). Testing for changes in trend in water quality data. PhD Faculty of Virginia


Jean-Pierre Barbot

Driss Boutat

T Floquet

An observation algorithm for nonlinear systems with unknown inputs

This paper provides some new developments in the design of unknown input observers for nonlinear systems. An algorithm which states if the state and the unknown input of the system can be recovered in finite time is introduced. This algorithm leads to the transformation of the system into an extended block triangular observable form suitable for the design of finite time observers. The proposed method is useful to relax some restrictive conditions of existing nonlinear unknown input observer design procedures.

Introduction

It is of importance to design observers for multivariable linear or nonlinear systems partially driven by unknown inputs. Such a problem arises in systems subject to disturbances or with inaccessible inputs and in many applications such as parameter identification, fault detection and isolation or cryptography. The design of observers for nonlinear systems is a challenging problem, even for accurately known systems. It has received a considerable amount of attention in the literature. In many approaches, nonlinear coordinate transformations are used to transform the system into suitable observer canonical forms. Then, observers with linearizable error dynamics (see e.g. [START_REF] Glumineau | New Algebro-Geometric Conditions for the Linearization by Input-Output Injection[END_REF][START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF][START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF]), high gain observers [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF] or backstepping observers [START_REF] Krener | Locally convergent nonlinear observers[END_REF] can be designed. Few works deal with the design of unknown input observers for nonlinear systems. Some of them are concerned with applications in the field of fault detection and identification, and in particular the nonlinear Fundamental Problem of Residual Generation [START_REF] Floquet | On the robust fault detection via a sliding mode disturbance observer[END_REF][START_REF] Hammouri | Observer-Based Approach to Fault Detection and Isolation for Nonlinear Systems[END_REF][START_REF] De Persis | A geometric approach to nonlinear fault detection and isolation[END_REF]. Other ones deal with nonlinear systems subject to exogenous perturbations and are based on sliding mode considerations [START_REF] Xiong | Sliding Mode Observer for Nonlinear Uncertain Systems[END_REF]. The convergence is obtained in finite time under the assumptions that the system can be put 1 into a set of triangular observable forms, where the unknown inputs act only on the last dynamics of each triangular form. This assumption is known as the observability matching condition. The present work aims at the development of a systematic method leading to the finite time observation of a class of nonlinear systems with unknown inputs even if the observability matching condition is not fulfilled. To this end, the procedure given in [START_REF] Floquet | An observability form for linear systems with unknown inputs[END_REF] is extended to the nonlinear case. It also results in a constructive algorithm that transforms the system into a similar type of block triangular observable forms. This transformation relies on the introduction of suitable fictitious outputs. Sufficient conditions for the existence of such auxiliary variables are given. After transformation of the system, it is shown that it is possible to design observers that provide the finite time estimation of both the state variables and the unknown inputs. An illustrative example highlights the efficiency of the proposed methodology.

Problem statement and motivations

Consider, on an open set U , the nonlinear system:

   ẋ = f (x) + g(x)w = f (x) + m i=1 g i (x)w i y = h(x) = [h 1 (x), .., h p (x)] T (1)
where x ∈ U ⊂ R n is the state vector, y ∈ R p is the output vector and where w = [w 1 , ..., w m ] ∈ R m represents the unknown inputs. The vector fields f and g 1 , ..., g m , and the functions h 1 , ..., h p , are assumed to be sufficiently smooth on U . Without loss of generality, it is assumed that p ≥ m, and that for all x ∈ U , the distribution G = span {g 1 , . . . , g m } and the codistribution span {dh 1 , . . . , dh p } are nonsingular on U . Like in [START_REF] Marino | Almost disturbance decoupling for single-input single-output nonlinear systems[END_REF], let us first define the unknown input characteristic indexes {ρ 1 , ..., ρ p } such that, for 1 ≤ i ≤ p:

L gj L k f h i (x) = 0, for k < ρ i -1
, and for all 1 ≤ j ≤ m, L gj L ρi-1 f h i (x) = 0, for at least one 1 ≤ j ≤ m, for all x ∈ U . The system (1) with w = 0 is supposed to be locally weakly observable on U [START_REF] Hermann | Nonlinear controllability and observability[END_REF]. Thus, there exists a change of coordinates

φ = h 1 , ..., L ν1-1 f h 1 , ..., h p , ..., L νp-1 f h p T
such that, after a suitable reordering of the state components, the system (1) is locally transformed into

ξi = A δi ξ i + H δi V δi (x, w) (2)
.

η = a(ξ, η) + b(ξ, η)w (3)
y i = C δi ξ i with ξ = ξ T 1 , ..., ξ T p T , ξ i ∈ R δi , 1 ≤ i ≤ p V δi (x, w) = L δi f h i (x) + m i=1 L gj L δi-1 f h i (x)w j ∈ R A δi =        0 1 0 0 0 0 0 1 0 0 . . . . . . . . . . . . . . . 0 0 0 0 1 0 0 0 0 0        ∈ R δi×δi H δi = 0 0 ... 1 T ∈ R δi , C δi = 1 0 ... 0 ∈ R 1×δi
and a, b are smooth vector fields on U . The integers (ν 1 , ν 2 , . . . , ν p ) are the so-called observability indices of system (1) (see [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] for a definition) and thus satisfy

ν 1 + • • • + ν p = n.
From the definition of the ρ i , one has δ i = min (ν i , ρ i ) and g δi = 0 if ν i < ρ i (i.e. δ i = ν i ).

A triangular observable form

Most existing nonlinear observers for system (1) are designed under the assumption that the system satisfies the so-called observability matching condition. This condition was first formulated for SISO linear systems in [START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF], Chapter 4. In the general case of MIMO nonlinear systems, a necessary and sufficient condition is given by:

ν i ≤ ρ i for all 1 ≤ i ≤ p. (4)
Then, δ i = ν i for all 1 ≤ i ≤ p, and under the change of coordinates ξ = φ(x) the system (1) is transformed into the form:

ξi = A νi ξ i + H νi V νi (ξ, w) (5)
y i = C νi ξ i with m i=1
L gj L νi-1 f h i (x)w j = 0 if and only if ν i = ρ i . Each subsystem of ( 5) is in the so-called triangular observable form. Finite time observers for such a form can be found in the literature, based on step-by-step sliding mode techniques [START_REF] Xiong | Sliding Mode Observer for Nonlinear Uncertain Systems[END_REF], higher order sliding modes [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF][START_REF] Levant | Robust Exact Differentiation via sliding mode technique[END_REF], numerical approaches [START_REF] Diop | On numerical differentiation algorithms for nonlinear estimation[END_REF][START_REF] Kang | Moving Horizon Numerical Observers of Nonlinear Control Systems[END_REF], or algebraic methods [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF]. The design of such observers is left to the reader, since they are straightforward applications of existing results. Nevertheless, depending on the choice of the observer, some assumptions have to be introduced. For instance, in all the previously mentioned works, the unknown input w has, at least, to be bounded. A necessary and sufficient condition for the recovery of the unknown inputs is that

Γ(x) =     L g1 L ρ1-1 f h 1 (x) ... L gm L ρ1-1 f h 1 (x) . . . . . . L g1 L ρp-1 f h p (x) ... L gm L ρp-1 f h p (x)    
has rank m. In this case, ρ = {ρ 1 , ..., ρ p } is the vector relative degree as defined in [START_REF] Isidori | Nonlinear Control Systems, Communication and Control Engineering Series[END_REF], p. 220, when m = p.

An extended triangular observable form

The aim of this paper is to provide an observation algorithm that allows for the finite time estimation of both the state and the unknown inputs of (1) even if ν j > ρ j for at least a j in {1, ..., p}. Consider again the general form [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF][START_REF] Diop | On numerical differentiation algorithms for nonlinear estimation[END_REF].

Applying any of the aforementioned finite time observers: (i) for 1 ≤ i ≤ p, ξ i can be estimated in finite time; (ii) one can also recover in finite time the last component V δi of each subsystem of ( 2). The problem is to recover the remaining state η. Denote:

V (x) =      V δ1 (x, w) V δ2 (x, w) . . . V δp (x, w)      =       L δ1 f h 1 (x) L δ2 f h 2 (x) . . . L δp f h p (x)       + Γ δ (x)w where Γ δ (x) =     L g1 L δ1-1 f h 1 (x) ... L gm L δ1-1 f h 1 (x) . . . . . . L g1 L δp-1 f h p (x) ... L gm L δp-1 f h p (x)     .
Let £ be the commutative algebra of the measured outputs and their successive Lie derivatives up to order δ i : £ = span{h 1 , ..., L δ1-1 f h 1 , ..., h p , ..., L δp-1 f h p } and let d£ be the codistribution:

d£ = span{dh 1 , ..., dL δ1-1 f h 1 , ..., dh p , ..., dL δp-1 f h p }.
Assume there exists a 1 × p row vector

K(x) = (k 1 (x), ..., k p (x)) = 0, k i ∈ £ for 1 ≤ i ≤ p, such that: K(x)Γ δ (x) = 0 for all x ∈ U (6)
and set:

ȳ = h(x) = K(x)V (x) = p i=1 k i (x)L δi f h i (x) (7)
Note that ȳ is an available information (after a finite time) and is not affected by the unknown inputs. Therefore, if d£+span d h d£, ȳ can be considered as an additional fictitious output. Then, let ρi and νi be the unknown input characteristic indexes and the observability indices of (1) with respect to the extended output y T , ȳ

T = h T , h(x)
T . If νi ≤ ρi for all 1 ≤ i ≤ p + 1, the system (1) can be transformed into:

ξi = A νi ξ i + H νi V νi ( ξ, w) for 1 ≤ i ≤ p y i = C νi ξ i ξ = A νp+1 ξ + H νp+1 V νp+1 ( ξ, w) ȳ = C νp+1 ξ where ξ = ξ T 1 , ..., ξ T p , ξT T ∈ R n .
Then, it is possible to recover both the state and the unknown inputs in finite time. Let us give sufficient conditions for the existence of a suitable fictitious output ȳ. For this, the following notations are introduced: i) G ⊥ = span{̟ 1 , ..., ̟ n-m }, the annihilator of G (̟ i are 1-forms such that ι g k ̟ i = 0, where ι g ̟ is the inner product of the vector field g and the 1-form ̟).

ii) Ω £ , the module spanned by d£ over £.

Proposition 1

The following conditions are equivalent:

i) Equation ( 6) has a solution K and KV / ∈ £.

ii

) Ξ = span{̟ ∈ G ⊥ ∩ Ω £ such that ι f ̟ / ∈ £} = {0}. Proof : Set ̟ = p i=1 k i dL δi-1 f h i with k i ∈ £. Clearly, ̟ ∈ Ω £ and ι f ̟ = ι f p i=1 k i dL δi-1 f h i = p i=1 k i L δi f h i = KV = ȳ. KΓ δ = K     dL δ1-1 f h 1 . . . dL δp-1 f h p     g 1 • • • g m = ̟. g 1 • • • g m Thus: {K is a solution of (6) such that KV / ∈ £} ⇐⇒ {̟ ∈ Ω £ , ι τ ̟ = 0 for any τ ∈ G and ι f ̟ / ∈ £} ⇐⇒ {̟ ∈ G ⊥ ∩ Ω £ and ι f ̟ / ∈ £} ⇐⇒ Ξ = {0}.
The discussion above can be recursively generalized as follows. Assume that the condition (4) is not still satisfied with the extended output obtained with the solutions of (6). On the basis of this new output, the corresponding matrix Γδ can be computed and another set of fictitious outputs can eventually be found. One can iterate this procedure until the condition ( 4) is fulfilled for a new extended output. Then, the original system can be put into an extended block triangular observable form 1 :

ξ1 i = A ν 1 i ξ 1 i + H ν 1 i V ν 1 i (ξ, w) y 1 i = y i = C ν 1 i ξ 1 i , 1 ≤ i ≤ p 1 ξ2 i = A ν 2 i ξ 2 i + H ν 2 i V ν 2 i (ξ, w) y 2 i = C ν 2 i ξ 2 i , 1 ≤ i ≤ p 2 . . . ξk * i = A ν k * i ξ k * i + H ν k * i V ν k * i (ξ, w) y k * i = C ν k * i ξ k * i , 1 ≤ i ≤ p k * ( 8
)
where the integers ν j i are the observability indices of the system (1) with the new outputs y j i . The first subsystem is fed by the original outputs of the system. A finite time observer is designed to estimate the state of this subsystem and to provide in finite time the knowledge of the fictitious outputs y 2 i , 1 ≤ i ≤ p 2 . Then, the state of the second triangular observable form can be estimated as well as the fictitious outputs y 3

i . Thus, one can recursively obtain the whole state of the system in finite time.

Remark 1

The finite time property is required to ensure that one obtains a fast and accurate estimation of the fictitious outputs (for instance, via the equivalent output injection in the case of sliding mode observers, see [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF] and the references therein). Furthermore, this property is often desirable in the framework of observation and particularly for the purpose of observer-based controller design for nonlinear systems. Then, for a large class of nonlinear systems, the observer can be designed separately from the controller and the separation principle does not need to be proved. It can also be of paramount importance in applications that require fast estimations of some unknown inputs like fault detection and identification or on-line parameter identification.

Nonlinear unknown input observer algorithm

An algorithm that states if the system can be transformed into [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF] and that provides the integers p j , ν j i and the auxiliary outputs y j i (j = 1, ..., k * ) is now given.

Step 0: Compute G and its annihilator G ⊥ . Set p 1 = p, h 1 1 , .., h 1

p 1 T = [h 1 , .., h p ] T , µ 0 = 0, z 0 1 = ... = z 0 µ 0 = 0. Step α: [a] Consider y α = h α 1 , .., h α p α
T ∈ R p α and reorder its components as follows:

y α = h α 1 , ..., h α l α , h α l α +1 , ..., h α p α T 1
Systems that admit such a form belong to the class of left invertible systems with trivial zero dynamics (see [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF]).

such that for 1 ≤ j ≤ l α :

∀i ∈ [1, . . . , m], ∀k ∈ N L gi L k f h α j = 0
and for 1 ≤ j ≤ p α -l α , there exists an integer ρ α j such that:

∀i ∈ [1, . . . , m] L gi L k f h α l α +j = 0 ∀k < ρ α j -1 ∃i ∈ [1, . . . , m] L gi L ρ α j -1 f h α l α +j = 0. [b] Define Φ α = {dh α 1 , ..., dL n-1 f h α 1 , ..., dh α l α , ..., dL n-1 f h α l α }. Compute I α = span dz α-1 1 , ..., dz α-1 µ α-1 Φ α .
Let dimI α = µ α-1 + ϕ α . I α can be written as follows:

I α = span{dz α-1 1 , ..., dz α-1 µ α-1 , dh α 1 , ., dL ϕ α 1 -1 f h α 1 , ., dh α l α , ., dL ϕ α l α -1 f h α l α } with l α i=1 ϕ α i = ϕ α . If µ α-1 + ϕ α = n, set µ k * = µ α-1 + ϕ α dz k * 1 , ..., dz k * µ k * = {dz α-1 1 , ..., dz α-1 µ α-1 , dh α 1 , ., dL ϕ α 1 -1 f h α 1 , ., dh α l α , ., dL ϕ α l α -1 f h α l α }
and stop the algorithm.

[c] If µ α-1 + ϕ α < n, consider the outputs affected by the unknown inputs and define: Υ α = {dh α l α +1 , ., dL

ρ α 1 -1 f h α l α +1
, ., dh α p α , ., dL

ρ α p α -l α -1 f h α p α } Compute the codistribution Ω α = span {I α Υ α }. Let dimΩ α = µ α-1 + ϕ α + κ α = µ α and write Ω α = span{dz α 1 , ..., dz α µ α } with {dz α 1 , ..., dz α µ α } = {dz α-1 1 , ..., dz α-1 µ α-1 , dh α 1 , ..., dL ϕ α 1 -1 f h α 1 , ..., dh α l α , ..., dL ϕ α l α -1 f h α l α , dh α l α +1 , ..., dL κ α 1 -1 f h α l α +1 , ..., dh α p α , ..., dL κ α p α -l α -1 f h α p α }.
and

p α -l α i=1 κ α i = κ α . If µ α = n, the algorithm stops. [d] Otherwise, µ α < n. Define £ α = span{z α 1 , ..., z α µ α } Ω α £ = span µ α i=1 φ i dz α i , φ i ∈ £ α Ξ α = span ̟ ∈ G ⊥ ∩ Ω α £ such that ι f ̟ / ∈ £ α .
Let p α+1 = dim Ξ α . If p α+1 = 0, the state of the system (1) can not be recovered with the method described in this paper and the algorithm stops. Otherwise, there exist p α+1 one-forms ̟ i such that Ξ α = span ̟ 1 , ., ̟ p α+1 and one can define the following vector of fictitious outputs, suitable to the problem (see Proposition 1):

y α+1 = ι f ̟ 1 , ..., ι f ̟ p α+1 T . Set h α+1 1 , .., h α+1 p α+1 T = ι f ̟ 1 , ..., ι f ̟ p α+1 T . Go to [a].
If the algorithm stops for some µ k * = n, the change of coordinates

φ = z k * 1 , ..., z k * µ k *
T is well defined and transforms the system into a set of block triangular observable forms similar to [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF]. The functions h i j , and the integers p i , l i , ϕ i j , κ i j are obtained in each i-th iteration of the algorithm.

Remark 2 If the condition ( 4) is satisfied for the measured outputs of the system (1), µ 1 = n after the first iteration of the algorithm and the system is exactly transformed into the form ( 5) that is usually considered for the design of asymptotic (see e.g. [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF][START_REF] Krener | Locally convergent nonlinear observers[END_REF]) or robust finite time ( [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF][START_REF] Xiong | Sliding Mode Observer for Nonlinear Uncertain Systems[END_REF]) nonlinear observers.

Estimation of the unknown inputs

If the algorithm ends in a positive way, the unknown inputs can also be obtained in a finite time. Indeed, the use of a finite time observer provides an estimation of the state, say x, and the knowledge of the following quantities (from the last line of each block of the triangular form):

θi j = L κ i j f h i l i +j (x) + m s=1 L gs L κ i j -1 f h i l i +j (x)w s , (9)
for 1 ≤ j ≤ p i -l i , and 1 ≤ i ≤ k * . The relations (9) can be rewritten as: Λ(x)w = Θ(x) with

Λ(x) =      Lg 1 L κ 1 1 -1 f h 1 l 1 +1 (x) ... L gm L κ 1 1 -1 f h 1 l 1 +1 (x) . . . . . . L g 1 L κ k * p k * -l * -1 f h k * p k * (x) ... L g m L κ k * p k * -l * -1 f h k * p k * (x)      Θ(x) =      θ1 1 -L κ 1 1 f h 1 l 1 +1 (x) . . . θk * p k * -l * -L κ k * p k * -l * f h k * p k * (x)     
Since the distribution span {g 1 , . . . , g m } is assumed to be nonsingular, the matrix

Λ(x) =      dL κ 1 1 -1 f h 1 l 1 +1 (x) . . . dL κ k * p k * -l * -1 f h k * p k * (x)      g 1 • • • g m
has rank m on every subset of U where at least m one-forms dL κ i j -1 f h i l i +j does not belong to G ⊥ . Thus, an estimation of the unknown input is given by: w = Λ + (x)Θ(x) where Λ + is a well defined pseudo-inverse of Λ.

Example

As a way of illustration2 , consider the following nonlinear system subject to the unknown input w = [w 1 , w 2 ]

T ẋ1 = x 2 -x 3 1 ẋ2 = x 3 + x 2 2 -x 3 2 + a(x 3 , x 4 )w 1 ẋ3 = x 5 (10) ẋ4
= -x 4 + x 2 2 + b(x 2 , x 3 )w 1 ẋ5 = -x 3 + x 2 w 2 ẋ6 = -x 6 + w 2
with outputs y 1 = x 1 , y 2 = x 4 and y 3 = x 6 . The scalar functions a and b are such that a(x 3 , x 4 ) = a 1 (x 3 )a 2 (x 4 ), b(x 2 , x 3 ) = a 1 (x 3 )b 2 (x 2 ) and where a(0, 0) = 0 and b(0, 0) = 0, and b 2 (x 2 ) = 0, ∀x 2 ∈ R. For this system, one has ρ 1 = 2, ρ 2 = ρ 3 = 1 and ν 1 = 4, ν 2 = ν 3 = 1 (as a consequence δ 1 = 2, δ 2 = δ 3 = 1). Thus, the necessary and sufficient condition (4) for a system to be transformed into a form similar to (5) is not fulfilled. Furthermore, the distribution G is not involutive. As a consequence, the sliding mode observer proposed in [START_REF] Xiong | Sliding Mode Observer for Nonlinear Uncertain Systems[END_REF], where the distribution spanned by the unknown input channels has to be involutive, can not be designed here. However, the procedure proposed in this paper is applicable. The annihilator of G is given by:

G ⊥ = span{dx 1 , dx 3 , b 2 dx 2 -a 2 dx 4 , dx 5 -x 2 dx 6 }
In the first step of the algorithm, I 1 is empty and

Ω 1 = span{dh 1 , dL f h 1 , dh 2 , dh 3 } = span{dx 1 , -3x 2 1 dx 1 + dx 2 , dx 4 , dx 6 } with dim Ω 1 = 4 < n. One has £ 1 = span{h 1 , L f h 1 , h 2 , h 3 } = span{x 1 , x 2 -x 3 1 , x 4 , x 6 } = span{x 1 , x 2 , x 4 , x 6 } and b 2 dx 2 -a 2 dx 4 ∈ Ω 1 £ . Then G ⊥ Ω 1 £ = span{dx 1 , b 2 dx 2 -a 2 dx 4 } Ξ 1 = span{b 2 dx 2 -a 2 dx 4 }.
Ξ 1 = 0 and one can define the following fictitious output:

y 2 = ι f (b 2 dx 2 -a 2 dx 4 ) = b 2 (x 3 + x 2 2 -x 3 2 ) -a 2 (-x 4 + x 2 2 ) = b 2 (x 2 )x 3 mod[£]
Since b 2 (x 2 ) = 0 for all x 2 , the knowledge of y 2 is equivalent to the knowledge of x 3 . The second step starts by defining the output vector: y 2 = x 3 . I 2 is empty and Ω 2 = span{dx 1 , dx 2 , dx 4 , dx 6 , dx 3 , dx 5 } with dim Ω 2 = 6. Then, one can define the change of coordinates: z = ψ(x) = [x 1 , x 2 , x 4 , x 6 , x 3 , x 5 ] T . In the new coordinates, the system is rewritten as:

ż1 = z 2 -z 3 1 ż2 = z 5 + z 2 2 -z 3 2 + a 1 (z 5 )a 2 (z 3 )w 1 ż3 = -z 3 + z 2 2 + a 1 (z 5 )b 2 (z 2 )w 1 (11) ż4
= -z 4 + w 2 ż5 = z 6 ż6 = -z 5 + z 2 w 2
By the means of a finite time observer fed by the original outputs of system [START_REF] Hammouri | Observer-Based Approach to Fault Detection and Isolation for Nonlinear Systems[END_REF], i.e. y 1 = z 1 , y 2 = z 3 and y 3 = z 4 , one can recover the state z 2 and V 1 = z 5 + z 2 2 -z 3 2 + a 1 (z 5 )a 2 (z 3 )w 1 V 2 = -z 3 + z 2 2 + a 1 (z 5 )b 2 (z 2 )w 1 V 3 = -z 4 + w 2

For the sake of place, the design of the observer is not given here. However, the reader can for instance refer to [START_REF] Floquet | A canonical form for the design of unknown input sliding mode observers[END_REF] where an example of higher order sliding mode observer, designed for a similar problem in the linear case, can be extended to the problem of nonlinear systems. Then,

y 2 = ι f (b 2 (z 2 )dz 2 -a 2 (z 3 )dz 3 ) = b 2 (z 2 ) z 5 + z 2 2 -z 3 2 + a 1 (z 5 )a 2 (z 3 )w 1 -a 2 (z 3 ) z 2 2 + a 1 (z 5 )b 2 (z 2 )w 1 = b 2 (z 2 )V 1 -a 2 (z 3 )V 2 = b 2 (z 2 )(z 5 + z 2 2 -z 3 2 ) -a 2 (z 3 )(-z 3 + z 2 2 )
is known after a finite time and z 5 =

y 2 +a2(z3)(-z3+z 2
2 ) b2(z2) + z 3 2 -z 2 2 is an available information. Again, a finite time observer leads to the recovery of z 6 . Then, the unknown input can also be obtained since: w 1 = V1-z5-z 2 2 +z 3 2 a1(z5)a2(z3) and w 2 = V 3 + z 4 .

Conclusion

In this paper, the problem of the observation of nonlinear systems subject to unknown inputs was considered. An observation algorithm that determines whether it is possible to recover the state and unknown input in finite time was introduced. When the answer is yes, the algorithm provides a change of coordinates that transforms the system in a new type of block triangular observable form well suited to the design of finite time observers. The observability matching condition usually required for the design of nonlinear unknown input observers is relaxed with the proposed method.

A physical application of the proposed algorithm in the field of chaotic synchronization for secure communication can be found in[START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF].