Adèle Bressy
email: adele.bressy@leesu.enpc.fr

Marie-Christine Gromaire
email: gromaire@leesu.enpc.fr

Catherine Lorgeoux
email: catherine.lorgeoux@g2r.uhp-nancy.fr

Mohamed Saad
email: mohamed.saad@leesu.enpc.fr

Florent Leroy
email: florent.leroy@hydra.setec.fr

Ghassan Chebbo
email: ghassan.chebbo@leesu.enpc.fr

Efficiency of source control systems for reducing runoff pollutant loads: Feedback on experimental catchments within Paris conurbation

Keywords: flooding source control system, frequent rain event, micropollutants, stormwater, SUDS

Highlights:

SUDS designed for peak flow control were monitored (flow, organic pollutants, metals) Volume and contaminant loads were reduced by all SUDS even for ordinary events Efficiency of SUDS depends strongly on the type of storage and its general conception

Strong reliance between pollutant mitigation and water volume reduction

Runoff reduction-oriented design of SUDS: an efficient solution for pollutant mitigation

Introduction

Stormwater management has become a critical issue in the field of sustainable urban development to protect civil society against flood and because runoff on urban surfaces has been recognised as a major cause of the degradation of receiving waters [START_REF] Burton | Stormwater Effects Handbook: A Toolbox for Watershed Managers[END_REF]). In the past, stormwater was collected by drainage networks, but with fast urbanization these networks have become inadequate, leading local authorities to develop strategies to prevent flooding.

The first strategy adopted was the large-scale management of urban drainage systems by building large reservoirs. It was not sufficient to remove the flooding risks and now a local stormwater management approach is preferred [START_REF] Brombach | A new database on urban runoff pollution: comparison of separate and combined sewer systems[END_REF][START_REF] Ellis | The management of urban surface water drainage in England and Wales[END_REF][START_REF] Jefferies | Sustainable Urban Drainage Systems (SUDS) treatment train assessment tool[END_REF][START_REF] Roy | Impediments and solutions to sustainable, watershedscale urban stormwater management: Lessons from Australia and the United States[END_REF]. In recently urbanised areas, facilities are developed simultaneously to the urban growth promoting retention or infiltration at a small scale. These facilities are often called "Sustainable Urban Drainage Systems" (or SUDS). Two major types of SUDS design are used worldwide: flow rate regulation and volume regulation. Both in France and in the USA, the most widespread regulation is based on a limited flow rate value [START_REF] Petrucci | Do stormwater source control policies deliver the right hydrologic outcomes[END_REF][START_REF] Roy | Impediments and solutions to sustainable, watershedscale urban stormwater management: Lessons from Australia and the United States[END_REF]. For example in the French department Seine Saint-3/43 Denis, in the suburb of Paris, the local authorities have imposed a flow rate regulation at 10 l/s/ha since 1993 [START_REF] Dea | Règlement de l'assainissement départemental. In Direction de l'Eau et de l'Assainissement de la Seine Saint-Denis[END_REF]. Thus SUDS are typically intended to facilitate hydraulic management and have been designed for exceptional precipitation events; only on rare occasion are contamination mitigation objectives actually addressed [START_REF] Martin | Urban stormwater drainage management: The development of a multicriteria decision aid approach for best management practices[END_REF].

Studies have revealed that such SUDS are capable of: reducing the discharged volumes, delaying catchment response, slowing flow velocities and increasing water residence time within the various facilities [START_REF] Jefferies | Assessing the performance of SUDS for the management and control of surface water runoff in Scotland[END_REF][START_REF] Scholes | A systematic approach for the comparative assessment of stormwater pollutant removal potentials[END_REF]. Thus they can have a substantial impact on the pollutant fluxes being conveyed by stormwater and discharged into receiving waters. Purifying effects have indeed been observed at the system scale for several types of SUDS [START_REF] Jefferies | Assessing the performance of SUDS for the management and control of surface water runoff in Scotland[END_REF][START_REF] Pagotto | Comparison of the hydraulic behaviour and the quality of highway runoff water according to the type of pavement[END_REF][START_REF] Vanwoert | Green roof stormwater retention: Effects of roof surface, slope, and media depth[END_REF]. However, there are few studies highlighting the overall effect of SUDS on pollutant fluxes control, at a suburban catchment scale. The effect of SUDS that were designed for flow control and not pollutant control remains poorly documented. Moreover literature data is usually limited to metals and nutrients and few data is available on organic micropollutants [START_REF] Diblasi | Removal and Fate of Polycyclic Aromatic Hydrocarbon Pollutants in an Urban Stormwater Bioretention Facility[END_REF][START_REF] Matamoros | Evaluation of the seasonal performance of a water reclamation pond-constructed wetland system for removing emerging contaminants[END_REF].

Therefore, the objective of this research is to assess the effect of peak flow control policies, on the water and contaminant flows discharged during frequent rain events at a small catchment scale. A special attention has been given to a selection of priority substances listed in the Water Framework Directive (2000/60/EC), whose presence is significant in runoff [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]), but whose fate in SUDS is not much documented to date. Three catchments containing SUDS were compared to a reference catchment featuring a conventional separate sewer network, in terms of hydraulic behaviour and discharged contaminant fluxes (i.e., suspended solids (SS), organic carbon (OC), trace metals (copper, lead, zinc) and organic micropollutants: polycyclic aromatic hydrocarbons (PAHs), polychlorinated biphenyls (PCBs), and alkylphenols). Moreover, the deposits formed in storage zones were characterised so as to better understand the fate of micropollutants during their transfer and in order to devise the best strategy for recovering and treating these wastes.

Materials and methods

Site characterisation

A residential site, characterised by low-density traffic and no industrial activity within a 5-km radius, was studied in a suburban area near Paris (France). The site was drained by a separate sewer system. Land use on this site was quite homogenous, while the stormwater management system featured a wide diversity.

On this site, four small catchments ranging from 0.8 ha to 1.9 ha were studied. The "Reference" catchment was drained by a conventional separate sewer system, while the other three catchments ("North", "Park" and "South") temporarily stored stormwater in various SUDS to comply with the 10 l/s/ha flow limitation imposed by local authorities. Stormwater on the North catchment was stored in a vegetated roof and in an underground pipeline for common rain events (up to 1 year return period) with an overflow onto a swale or on parking for exceptional events. In the Park catchment, stormwater was stored in a grassed detention pond that is part of a public garden. Stormwater management on the South catchment had been incorporated into the land use plan and the practices associated various types of storage facilities: underground tank for private parcels, swales and a public square covered by grass.

The outlets of the catchments with SUDS are fitted with flow-rate regulators as usual in France (Table 1). According to [START_REF] Martin | Urban stormwater drainage management: The development of a multicriteria decision aid approach for best management practices[END_REF], these SUDS were representative of the kinds of solutions adopted in France.

The characteristics of the catchments are listed in Table 1. The four catchments displayed a homogeneous pattern of urbanisation and were located adjacent to one another (less than contributions (i.e. rainfall and deposits). Differences in land use breakdown appeared across these four catchments. The breakdown of the North catchment was similar to the Reference, though automobile traffic was heavier on North due to the presence of retail shops. The Park catchment was mainly composed of buildings and gardens and contains no streets. The South catchment was relatively devoid of streets and contained a higher density of pedestrian paths than the Reference. Consequently, the discharges for these four catchments could not be directly compared and required introducing a set of land use-based modelling tools.

With the objective of establishing a model, the potential contaminant entry paths (atmospheric fallout, pavement runoff and runoff from built parcels) were also examined. Atmospheric fallout was measured on the flat roof of the highest building within the study area. Both types of runoff (street and built parcel, i.e., roof and a private garden above slab underground parking) were evaluated on the Reference catchment. The characteristics of these two subcatchments are provided in Table 1 and described with greater details in [START_REF] Bressy | Alkylphenols in atmospheric depositions and urban runoff[END_REF].

Rainfall and flow measurements

Rainfall and flows at the four catchment outlets were continuously measured (every 0.2 mm for the rain and every minute for the flow) for one year between July 2008 and August 2009.

Instrumentation

Rainfall depth was recorded using a rain gauge (3029, Alcyr) placed on a flat roof in the study area. Runoff flows were measured at the Reference catchment outlet with a Sigma 950 flowmeter (water depth with a bubble pipe and velocity by Doppler Effect). At the sourcecontrolled catchment outlets, runoff flows were measured just beyond the flow regulation device with Sigma 950 flow-meters, by recording the water depth upstream of a V-notch weir.

Definition of a rain event

A rain event was defined as any precipitation leading to a flow signal. The beginning of an event was the time of the first precipitation data point during the 20 minutes preceding initiation of the flow signal for the Reference site and during the preceding 3 hours for the other three sites. The end of the event was defined as the time of the last precipitation data point during the flow signal period.

The following parameters were determined for each rain event:

-Peak flow: Q max (in l/s/ha) was the maximum flow value during the rain event;

-Lag time: T lag (h) was the time delay between maximum precipitation intensity and peak flow signal; -Emptying time: T empty (h) was the time delay between the end of precipitation and the end of the flow signal; -Runoff water depth: H runoff (mm) was the effective water depth discharged by the catchment during the rain event.

Sampling protocol and analytical procedure

Water sampling protocol

Both dry and wet bulk atmospheric depositions were sampled using 20-L bottles hermetically connected to a 1-m² stainless pyramidal funnel. The bottles were placed underneath the funnel just before the rain event and removed just afterward; they collected the wet deposition and the washoff of the contaminants deposited on the funnel during the previous dry weather period. Stormwater was collected from the storm sewer at the catchment outlet using automatic samplers (Bühler 1029) controlled via the flow meter. The sampling protocol was flow-proportional so as to obtain average concentrations throughout the event.

The campaigns conducted in order to analyse both organic contaminants (requiring the use of glass bottles) and metals (plastic bottles) were based on different sets of events. SS and organic matter were measured for all the events in glass or plastic bottles, but in this paper only SS and TOC data for rain events sampled simultaneously on atmospheric fallout, built parcel and street catchments were used. Table 2 provides the characteristics of the rain events considered for each parameter.

Soil and sediment sampling protocol

Sediment deposits were observed and sampled both in the North catchment storage pipe and upstream of the South catchment regulator. Several samples were collected in order to constitute a representative average sample of the sediment deposit. Average samples were reduced by quartering steps after homogenization. The soil of the public garden used as storage on Park catchment was also sampled. The retention basin surface was divided into three areas according to flooding frequency: one flooded at each rain event, another occasionally flooded and the last area was very seldom flooded. To constitute the average soil sample, 4 to 6 samples were collected in each area using a (20-cm long) corer, in following a 7-m 2 mesh grid pattern, and then combined. One sample was analysed in triplicate for a variability assessment: the signal deviation was below 20%.

Micropollutant analysis

The analytical procedures applied for organic compounds were previously described by [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]. Briefly, it was based on separating the dissolved and particulate fractions (threshold: 0.45 µm). Dissolved fraction was extracted on a SPE C18 cartridge, while a microwave-assisted extraction procedure was applied to the particulate fraction. The three pollutant families (PCBs, PAHs and alkylphenols) were then separated during a purification step on silica columns. Contaminants were quantified by internal calibration using gas chromatography coupled with mass spectrometry (GC/MS, Focus DSQ, ThermoFisher Scientific). Results are displayed as the sum of 13 PAHs3 (deriving from the US EPA list, excluding naphthalene, acenaphthene and acenaphthylene, which are too volatile to be correctly quantified), along with the sum of the 7 PCB indicators4 . Among alkylphenols, nonylphenols (NPs) and octylphenols (OPs) were studied.

Trace metals were analysed in both the total and dissolved fractions. Raw samples were microwave acid-digested at 95°C with nitric and hydrochloric acids. Filtered samples on 0.45 m cellulose acetate membranes were acidified to pH 1 with nitric acid. Metal concentrations were determined using Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES, Varian Vista MPX) through external calibration with a multielement standard solution (PlasmaNorm Multi-Elements).

The analytical uncertainties and the detection limits are given in Table S.1 in supplementary files.

Methodology used for site comparisons

The various land use breakdowns did not exhibit the same runoff coefficient and moreover did not produce the same pollutant quantities. Consequently a direct comparison of water and contaminant fluxes between the Reference site and sites equipped with SUDS proved to be an impossible task.

A water and contaminant emissions model was developed for each type of land use relative to the Reference catchment. Then this model was applied to the land use breakdowns of each catchment equipped with SUDS in order to simulate what the catchment behaviour would have been in the absence of flow regulation. This methodology is shown Fig. 1 andexplained below. It is important to note that our methodology did not aim an accurate simulation of the pollutant masses but the estimation of SUDS effectiveness. For this purpose, the pollutant masses were simulated using low assumptions for sources, which allows to be sure that SUDS effect on pollutants is not overrated.

Tool for runoff volume simulation

The aim of this model was to simulate the runoff volume that would have been produced with conventional drainage systems based on the rainfall depth and land use dataset. The model was adapted from [START_REF] Berthier | Simulation of regular rain events: the limits of the traditionnal initial losses and runoff coefficient scheme[END_REF] and distinguished 3 types of surfaces: roofs, streets and gardens.

The initial losses on roofs and streets were modelled by a surface storage depth IL (mm). The filling level of this surface storage at the beginning of a rain event depended on the amount of rainfall during the previous 3 hours (H 3h in mm) for roofs and previous 6 hours (H 6h in mm) for streets. Infiltration through the street's pavement was modelled using a constant infiltration rate K inf,s (in mm/h). If H is the rainfall depth (mm), T rain the rain event duration (h) and A r and A s the proportions of roof and street surface areas on the catchment, the runoff water depth (H runoff in mm) was given by: For roof runoff:

{ } [ ] r h r r runoff A H IL H H . 0 ; max 3 , - - = Equation 1
For street runoff:

If h s H IL 6 > , then: { } s rain s h s s runoff A T K H IL H H . 0 ; . ) ( max inf, 6 , - - - = Equation 2 otherwise: { } s rain s s runoff A T K H H . 0 ; . max inf, , - =
Private gardens above underground parking (50 to 100 cm soil) with drainage systems were modelled as a storage depth IL g (mm), with their filling level at the beginning of the event being dependent on rainfall quantity over the previous 6 days (H 6d in h). The storage filling rate during the rain event was modelled by a constant infiltration rate K inf,g ; evapotranspiration was not included in this modelling set-up. If A g is the proportion of gardens covering the catchment, the runoff water depth from private gardens (H runoff,g in mm) was given by:

If d g H IL 6 > , then: { } { } g rain g d g g runoff A T K H IL H H . 0 ; . ; min max inf, 6 , - - = Equation 3 otherwise: g g runoff A H H . , =
The runoff water depth from the public garden was assumed to equal zero because, in a conventional system, it would not have been connected to the sewer system.

The five model parameters (IL r , IL s , IL g , K inf,s and K inf,g ) were calibrated using rainfall and flow data over a 12-month period from the Reference catchment, by means of minimising the sum of absolute error values. The calibrated parameter values listed in Table 3 lie within the same interval as those found in the literature [START_REF] Berthier | Simulation of regular rain events: the limits of the traditionnal initial losses and runoff coefficient scheme[END_REF].

Tool for micropollutant simulation

The objective of this tool was to simulate contaminant emissions from catchments equipped with SUDS as if these catchments were being drained with a conventional sewerage system.

The principle consisted, for the sum of sampled events i (Cf. Table 2), of comparing the mass measured at the outlet ( M ) with the mass simulated by summing the masses input via the various entry paths ( M ) according to the equations 4 and 5. The validation of the tool was done with the Reference data.

∑ ⋅ = i i i Ref V C M , Equation 4 [ ] ∑ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ = + + = i g i g r i r i build s i s i s p i s i atm i building street path S CR S CR C S CR C S CR C H M M M M ) ( , , , , , , ,
Where, for the event i, i H is the rainfall depth; For zinc introduced via roofing materials, the corrosion models described in Gromaire et al.

(2011) were used, distinguishing the zinc roofs of other:

( )

∑             + ⋅ ⋅ ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ = + + + = ≠ = ≠ i i corrosion s i s i s i tot zinc r i r g i g p i s i atm i tot zinc building zinc building street path M S CR C H S CR S CR S CR C H M M M M M , , , , , , , , ,

Equation 5

Where, for the event i, To avoid overestimating the simulated masses when data from Reference were missing due to technical problems, i.e., to avoid overvaluing the SUDS effect during comparisons with measurements, the simulated values were deliberately minimised by adopting hypotheses based on entry path concentrations and runoff volumes. Only atmospheric input was considered for PCB assuming the recent buildings or cars do not release them. For PAHs and alkylphenols, when concentrations from road runoff were missing, the lowest measured value was used. Uncertainties on model results induced by input data uncertainty (especially measurement uncertainty on experimental data) were estimated with the law of propagation of uncertainties (explanation in supplementary files Annexe A.1).

Results and discussion

Performance of the simulation tools

The errors and relative errors between simulated and measured H runoff values are shown in Figure 2. Over a one-year period, model behaviour proved to be satisfactory (0.1% error between simulated and measured annual volumes) since the simulations have yielded good 12/43 results for the events producing the majority of yearly discharged water volume (60% of events were simulated with a margin of error less than ±30%, representing 80% of total annual rainfall). To minimise errors, this model was always applied to the sum of studied events, i.e., over the year for hydraulic simulation and over the sampled events for micropollutant simulation.

Table 4 offers a comparison, for the Reference catchment, of measured mass vs. simulated mass when aggregated over all sampled events (using Table 2 data). The simulated values lie within the uncertainty of the corresponding measurement values (uncertainty on both water volume and concentration measurements). This uncertainty does not cover scenario uncertainty according to the classification described by [START_REF] Warmink | Identification and classification of uncertainties in the application of environmental models[END_REF] since the scenario with the lowest value was initially chosen for sources. Our goal was not to develop a model to simulate actual masses of pollutant but to assess the SUDS effect. This choice allowed us to minimize the measured masses and therefore to avoid overestimation of the SUDS efficiency.

Effects of source control systems on discharged water

Flow dynamics at the event scale

Flow dynamics were studied for all rain events between July 2008 and August 2009. Over this period, 140 events could be distinguished on the Reference catchment (100 on North, 77 on Park and 114 on South). The difference in number of events across catchments was due to the slower dynamics of source-controlled catchments for which one event might correspond to several for Reference. These results have been analysed with a focus on frequent rain events, which represent most part of annual runoff volumes.

Peak flow reduction (see Fig. 3):

For Reference, Q max spanned a wide range of values (0.2 to 209 l/s/ha as the 1 st and 9 th deciles), yet it remained below 10 l/s/ha for 69% of rain events suggesting that for the majority of rain events the 10 l/s/ha regulatory flow threshold did not necessarily imply water retention. The situation would be quite different with lower thresholds: for instance, the 2 l/s/ha level would be exceeded for 66% of events.

Figure 3 provides a peak flow reduction for all source-controlled catchments. This effect was observed for almost all rain events, even ordinary rainfall episodes whose flow did not reach the nominal regulator flow (10 l/s/ha). The flow rate actually exceeded 2 l/s/ha for 19% and 13% of events at the North and Park outlets. At the South outlet, flow exceeded 2 l/s/ha for 63% of events; consequently, source control systems in place at the South site caused less impact on frequent rain events.

In the case of North catchment, peak flows were mainly controlled by the flow regulator.

However, for Park and South catchments, the level required for the initiation of the regulator was not reached for most rain event and peak flow attenuation was due to the natural retention time in the garden and in the swales.

Staggering and lag time of discharged water (see Fig. 4):

For the Reference catchment, flow dynamics were close to the rainfall dynamics: T lag varied between 3 and 14 minutes (1 st -9 th deciles), and T empty ranged from 0.8 to 5.8 hours. These high T empty values were due to drainage in the private gardens above underground parking.

For source-controlled catchments, the peak flows shifted in time relative to the rain peak: T lag varied from 16 minutes to 2.3 hours for North, 23 minutes to 2.6 hours for Park, and 8 minutes to 1.4 hours for South (1 st -9 th deciles).

14/43

The North and Park catchments also showed much longer emptying times for the aggregate of all rain events than either Reference or South. Storage emptying lasted between 1.7 and 17 hours for North and 2.3 to 10 hours for Park. The South catchment took between 0.9 and 3 hours to empty, which was of the same order of magnitude as Reference. The reactivity of the South site could be explained by the type of regulation system installed (vortex regulator, pump) with which flows quickly reached the nominal regulator flow. It may induce fewer effects when a small quantity of water is being stored, i.e., for ordinary rain.

These results indicate that the design of retention devices and, more importantly, the choice of regulation system have proven to be determinant as regards flow dynamics.

Effect on discharged volumes

For the three source-controlled catchments, the annual discharged volumes that would be generated with a conventional storm sewerage system were simulated with the model described in the section 2.4.1 and then compared in Table 5 to actual measured volumes as regards annual runoff coefficient (calculated as the ratio of rainfall amount to runoff water depth).

The annual runoff coefficient simulated for a conventional sewerage system was very close to the proportion of impervious surfaces for all 4 catchments. A large reduction (43% to 55%) in annual runoff volumes, compared to the hypothetical volumes with a conventional storm sewer, was observed for all 3 source-controlled catchments. This significant reduction was not surprising for the Park catchment since all its stormwater flowed through a garden and potentially infiltrated into the soil. For South, the reduction was correlated with longer and more extensive contact with vegetated surfaces (grass strips, swales, grassed basin). The reduction was much less expected on the North site, where storage facilities are mostly composed of impervious materials. One part of this reduction could be explained by vegetated roof and the other part by greater initial losses within the source control systems (dead volumes).

These reductions in annual runoff volumes represent very promising developments for stormwater quality management. Discharged masses are indeed partly determined by discharged volumes, as already observed by other authors [START_REF] Davis | Bioretention Technology: Overview of Current Practice and Future Needs[END_REF][START_REF] Trowsdale | Urban stormwater treatment using bioretention[END_REF], and lower annual runoff volumes can also induce a reduction in the discharged pollutant load.

Effects of source control systems on pollutant loads in stormwater

Figures 5a, 5c and 5e display, for all contaminants studied, both the masses measured (grey histogram) at the outfall of each catchment and the simulated masses (white histogram) as if the storm drainage system was conventional, for the sum of all monitored events. Water volume data have been added for comparison. Figures 5b,5d and 5f present the same types of data for average concentrations. Uncertainty bars associated with measured data correspond to measurement uncertainties and those associated with simulated data are model uncertainties (with a 80 % confidence interval) (explanation in supplementary files Annexe A.1).

North catchment

For the North catchment, the mass values simulated for a situation with no SUDS ever installed were higher than the measured masses on the whole, a finding that reveals a drop in onsite contamination. The effect on contaminant concentrations depended on the type of substance. Three categories of behaviour could be distinguished:

-For SS, total PAHs and zinc, the decrease in contaminant mass (50%, 60% and 72%, respectively) was greater than the drop in water volume (43%). The measured concentrations were thus lower than the simulation results. Since SS and PAHs are both particulate, we assumed the level of settling would be substantial in the underground storage zones, given that these decreases amounted to the same order of magnitude as the mass reductions evaluated for large storage basins [START_REF] Aires | Dépollution des polluants urbains de temps de pluie en bassin de stockage-décantation[END_REF][START_REF] Calabro | Simulation of the operation of detention tanks[END_REF][START_REF] Clark | Targeting treatment technologies to address specific stormwater pollutants and numeric discharge limits[END_REF]. The introduction of zinc, on the North site, occurred mainly in dissolved form via the corrosion of metallic roof materials (80% in dissolved form according to [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]). At the outfalls of large catchments, zinc has been proved to be 50% bind to particulate matter (Zgheib et al., 2011), which proves that zinc tends to bond with particles. It is suggested that part of the dissolved zinc became attached to particles and settled with them or else bonded with either the drainage system or deposits in the storage zone.

-For PCBs and NPs, in their total and dissolved form, the decrease in contaminant mass (between 24% and 36%) was assumed equal to the drop in water volume since the simulated concentrations lied within the uncertainty interval of measured concentrations. These mass amounts were thus lowered by the presence of storage, yet at a constant concentration. Let's recall that these substances were at around 70% in dissolved form in our samples [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]. This speciation was not expected to differ like it did for zinc since the distribution here is the one measured in stormwater [START_REF] Zgheib | Partition of pollution between dissolved and particulate phases: What about emerging substances in urban stormwater catchments?[END_REF] or for the natural environment (Cailleaud et al., 2007). It is therefore likely that a portion of the contamination has been trapped during water losses due to sedimentation / filtration for the particulate fraction and to adsorption / infiltration for the dissolved fraction.

-For the total (TOC) and dissolved organic (DOC) carbon, copper, dissolved PAHs and both total and dissolved OPs, the decrease in contaminant mass when assuming no SUDS had been installed was less than the drop in water volume or lied within the uncertainty interval of the measures. The mass of released contaminants was in fact lower by use of on-site storage, although the concentration released was slightly superior to that simulated for conventional sewer. These substances are in the both fractions [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF] and should therefore undergo at least the same decrease as the other substances, i.e. by sedimentation for the particulate fraction and adsorption for the dissolved fraction. One hypothesis for these findings might be that our simulation has underestimated the masses of these substances, as automotive traffic is a major source of copper, PAHs and OPs [START_REF] Bjorklund | Phthalates and nonylphenols in urban runoff: Occurrence, distribution and area emission factors[END_REF][START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF][START_REF] Motelay-Massei | Mass balance for polycyclic aromatic hydrocarbons in the urban watershed of Le Havre (France): Transport and fate of PAHs from the atmosphere to the outlet[END_REF]. Automotive traffic is more intense on the North catchment than the Reference site, which was used to calibrate the street-based contaminant production function. But as explained in paragraph 3.1, our model intentionally underestimates the simulated masses in order to avoid overestimation of the SUDS effect. As a consequence, for these substances, our methodology does not allow us to conclude about the effectiveness of SUDS. -For zinc, dissolved PAHs, total and dissolved NPs, and total and dissolved OPs, the decrease in contaminant mass (80%, 71%, 70% and 60%, respectively) was greater than or equal to the loss of water volume (60%). For these substances, which are mainly in dissolved form (80% for zinc, 79% for NP and 74% for OP according to [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]), an adsorption effect in the public garden was to be assumed, as demonstrated by [START_REF] Ray | Removal of selected pollutants from aqueous media by hardwood mulch[END_REF] on tree bark samples or by [START_REF] Scholes | A systematic approach for the comparative assessment of stormwater pollutant removal potentials[END_REF].

Park catchment

-For TOC, DOC, PCBs, dissolved PCBs and copper, the contaminant mass decrease was less than the actual volume loss. The simulated mass exceeded measurement results by respectively 29%, 29%, 42%, 50% and 22%, with measured concentrations topping the simulated concentration values. It was possible that the adsorption effect for these substances, which were at 57% in particulate fraction for TOC, 36% for PCBs and 72% for copper [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF], was less pronounced than for the group of substances described above.

-The mass of SS and total PAHs did not appear to be reduced by the upstream management systems introduced. The measured masses of these substances exceeded the simulated values, as if the fact of regulating discharges were raising the level of water contamination for these parameters. It was likely that our simulation has underestimated the actual SS mass produced, given that this value did not take into account particle production from the garden and playground (sandpit). As above, for these substances, our methodology does not allow us to conclude about the SUDS efficiency.

On this site, the decrease in water volume was high (over 50%), and the water residence time in the basins was quite long according to Section 3.2.1 (median drainage time exceeds 5 hours). The contact time between water and potential substrates (plants and soil particles) was also increased, thereby promoting both the adsorption of dissolved contaminants and their infiltration into the soil. Moreover, since the storage facilities were not enclosed, it was considered likely that the phenomenon of volatilisation, photolysis and biodegradation eliminated a portion of soil contamination during dry weather periods [START_REF] Scholes | A systematic approach for the comparative assessment of stormwater pollutant removal potentials[END_REF][START_REF] Weiss | Cost and pollutant removal of storm-water treatment practices[END_REF].

South catchment

On the South catchment, all simulated mass values exceeded measurements, which definitely points to contaminant interception within the various SUDS systems. This decrease in mass release however remains small in magnitude given that all simulated concentrations was less than or equal to the measured concentrations, with the exception of zinc.

-As regards zinc, the mass decrease (60%) exceeded the drop in water volume (46%), and the simulated zinc concentration was 24% higher than the measured value. On this catchment, the majority of zinc (90%) entered in dissolved form through the corrosion of roofing materials [START_REF] Gromaire | Zn and Pb emissions from roofing materials -modelling and mass balance attempt at the scale of a small urban catchment[END_REF] and the roof runoff was recovered in both an underground tank and planted swales. In these storage zones, sorption may indeed occur.

-As regards DOC, dissolved PCBs, dissolved PAHs, total and dissolved NPs, total and dissolved OPs and copper, the mass decline was similar to the reduction in water volume.

Since all these parameters were essentially dissolved [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF], sorption or infiltration effects were clearly apparent.

-The decreases in SS mass (23%), TOC (33%), total PCBs (15%) and total PAHs (19%) were all less than the loss of water volume (46%): these parameters tend to be more particulate in nature.

The decrease in South catchment contaminant mass thus appeared to be less pronounced than that of the other sites. Flow control systems on this catchment were less efficient on current flow rates than the other studied sites (Section 3.2.1), which results in lower residence times and hence worse efficiency.

Fate of micropollutants in these systems

Level of deposit contamination:

Figure 6 presents the contaminant contents in deposits at both the North (underground pipe) and South (storage zone upstream of the regulator) catchments.

The average PCB contents varied between 0.034 µg/g.dw for South and 0.058 µg/g.dw for North: this range was 3 times weaker than contents measured in the Reference stormwater SS (0.10 µg/g.dw, according to [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]). The particles held in storage prove to be the coarsest as well as the least contaminated; in addition, they are comparable to those detected by [START_REF] Jartun | Runoff of particle bound pollutants from urban impervious surfaces studied by analysis of sediments from stormwater traps[END_REF] in sediments from a separate urban sewer system in Norway (0.029 µg/g.dw) and below the limit established by the French decree relative to the spreading of sewage sludge (Decree No. 97-1133[START_REF] Dea | Règlement de l'assainissement départemental. In Direction de l'Eau et de l'Assainissement de la Seine Saint-Denis[END_REF], i.e. 0.8 µg/g.dw.

Average PAH contents was equal to 6.9 µg/g.dw in South sediment and 7.0 µg/g.dw in North sediment, which places them at roughly 5 times less than the contents measured in the Reference SS (33 µg/g.dw). These results were comparable to the values measured by [START_REF] Gasperi | Hydrocarbon loads from street cleaning practices: Comparison with dry and wet weather flows in a parisian combined sewer system[END_REF] in particles from water used for street cleaning and above the contents recorded by [START_REF] Jartun | Runoff of particle bound pollutants from urban impervious surfaces studied by analysis of sediments from stormwater traps[END_REF]: 3.4 µg/g.dw. The contents were between 3.5 and 4 times weaker than the limits established for the spreading of sewage sludge.

NPs contents ranged from 0.26 µg/g.dw at South to 6.3 µg/g.dw at North (0.04 and 0.70 µg/g.dw for OPs, respectively). The differences identified between these two sites may be explained by a smaller proportion of road on South and a building age effect subsequent to the European NP use restriction Directive (Directive 2003/53/EC). More specifically, the South catchment contained more recent construction, meaning that the materials employed could contain less NP. The contents found in Reference SS were measured at 6.8 µg/g.dw (0.27 µg/g.dw for OPs). [START_REF] Bjorklund | Phthalates and nonylphenols in urban runoff: Occurrence, distribution and area emission factors[END_REF] found lower contents in sediments from a separate sewer network in Sweden: below 1.5 µg/g.dw for NPs, and below detection limits for OPs. No regulation has been adopted for alkylphenols as regards their spreading.

For trace metals, the contents recorded at South revealed: 0.059 mg/g.dw copper, 0.041 mg/g.dw lead, and 2.8 mg/g.dw zinc. North catchment results yielded: 0.16 mg/g.dw copper, 0.12 mg/g.dw lead, and 0.79 mg/g.dw zinc. The higher copper and lead contents found in North stemmed from the greater volume of road traffic. The lower zinc values in North were due to a much larger proportion of zinc roofs in South (Table 1). These entire values were lower than those for Reference particles: 0.28 mg/g.dw copper, 0.26 mg/g.dw lead, and 5.5 mg/g.dw zinc. The order of magnitude remained the same as for measurements conducted by [START_REF] Jartun | Runoff of particle bound pollutants from urban impervious surfaces studied by analysis of sediments from stormwater traps[END_REF], i.e., 6 times less than the spreading limits for copper, 7 times less than those for lead and 4 times for zinc. These deposits were thus only slightly contaminated, especially when compared to the concentrations measured in stormwater SS: the coarsest, and hence least contaminated, particles are those retained in the SUDS. This category of particles does not require any special treatment, as opposed to the sludge generated by combined sewer networks.

Impacts of urban runoff retention on soil contamination:

To understand the fate of contaminants and to assess whether stormwater storage actually degrades garden soil quality in Park catchment or not, contaminant contents were measured in the soil as a function of the flooding gradient. Figure 7 displays the contents recorded across the various zones. OPs are not displayed because contents were around the LOQ.

PAH concentrations varied between 0.33 and 1.1 µg/g.dw, while PCB contents ranged from 0.026 to 0.060 µg/g.dw and NP values from 0.22 to 0.45 µg/g.dw. These PAH and PCB contents are comparable to measurements recorded in Seine River basin soils by [START_REF] Motelay-Massei | Distribution and spatial trends of PAHs and PCBs in soils in the Seine River basin, France[END_REF] and are below the values from the Canadian Soil Quality Guidelines (CCME, 2007) and far from the intervention values for remediation given by the regulation of Netherlands [START_REF] Vrom | Intervention values and target values -soil quality standards in Netherlands[END_REF]. In contrast, the garden soil seems to be contaminated by NPs, given the findings of [START_REF] Vikelsoe | Phthalates and nonylphenols in profiles of differently dressed soils[END_REF], who reported contents equal to 0.034 µg/g.dw in runoff storage areas, but this value stay below the Canadian Soil Quality Guidelines (CCME, 2007).

Copper contents varied between 0.014 and 0.040 mg/g.dw, lead between 0.020 and 0.083 mg/g.dw, and zinc between 0.056 and 0.10 mg/g.dw. These values are of the same order of magnitude as Paris region soils, according to Thévenot et al. (2007) and they respect guidelines (CCME, 2007;[START_REF] Vrom | Intervention values and target values -soil quality standards in Netherlands[END_REF].

A comparison across the 3 zones of increasing flood frequency did not reveal any difference in content for PCBs, and NPs. For PAHs and 3 trace metals, however, the most commonly flooded zones showed contents of between 1.3 and 3.2 times less. These differences may be attributed to sol heterogeneity or to an infiltration or degradation process, depending on contaminants. Moreover, these results demonstrate that the Park catchment soil is not significantly contaminated following its use for storage.

Discussion on micropollutant reduction measures for stormwater

Section 3.2 highlighted: i) a significant reduction in annual runoff volumes at the outlets of the source-controlled catchments due to losses on permeable surfaces and higher initial losses on impervious surfaces and ii) an increase in water retention time, which strongly depend on facilities design. When the design of SUDS considers only exceptional rainfalls, retention efficiency of common rainfall is highly variable depending on the type of storage (permeable or impervious surfaces) and the method of flow regulation. For the facilities that naturally increase the retention time, like the large grassed detention garden in Park catchment, the effects due to the flow regulator itself become insignificant for common rainfall. Section 3.3 proved that the presence of SUDS considerably lowered the contaminant masses released by the catchments. This conclusion was based on a comparison of fluxes measured with SUDS and simulated without SUDS. This methodology takes into account the measurement uncertainty. Scenario uncertainty was not evaluated, however the simulated scenario is an underestimating scenario. It is intentionally based on minimal emissions from the different pollutant sources, so as to avoid overestimation of the SUDS effectiveness.

This decrease varied from 20% to 80% of the total released mass, depending on both the study site and the type of pollutant. For underground storage facilities, the drop is substantial for particulate pollutants (SS, particle-bound contaminants) which are sensitive to sedimentation, and less pronounced for dissolved contaminants which might adsorb to the structures or deposits. For storage scenarios in permeable planted zones, the most significant contaminant decreases involve dissolved pollutants. Other effect of the SUDS can be observed on the substances primarily introduced into stormwater in dissolved form (i.e. metals from the corrosion of roofing materials) and that exhibits a strong affinity for organic matter. In this case, the binding of these substances to deposits and soil is assumed, inducing a stronger reduction of their release. Our findings should apply to other sites equipped with SUDS if they are designed with the same criteria. These results are interesting at local scale because it provides original data on organic micropollutants in SUDS and particularly useful at global scale because they allow to make effective recommendations for design criteria in terms of reducing pollution.

The effects on discharged contaminants may be correlated with hydraulic effects, namely the followings.

-Discharged masses are strongly correlated to discharged volumes and a reduction in annual runoff volumes may explain the highlighted mass decreases. Consequently, the volume drop for ordinary rain events needs to be assigned a top priority when developing stormwater management processes and, moreover, can be maximised by adding storages in grassy areas.

-During runoff, the reduction in flow velocity may lower contaminant erosion at the surface.

-In water storage facilities, reduced flow velocity allows for particle settlement [START_REF] Calabro | Simulation of the operation of detention tanks[END_REF]. A longer residence time increases contact time between water and substrate (i.e., soil, sediments, building materials) and is therefore favourable to both pollutant filtration trough permeable materials and sorption of dissolved fractions [START_REF] Mason | Behavior of heavy metals, nutrients, and major components during roof runoff infiltration[END_REF][START_REF] Ray | Removal of selected pollutants from aqueous media by hardwood mulch[END_REF].

-On a long-term basis and during dry weather periods, some of the pollutants retained in the source control devices may undergo evaporation or degradation (biodegradation, photodegradation, etc.) [START_REF] Scholes | A systematic approach for the comparative assessment of stormwater pollutant removal potentials[END_REF][START_REF] Warren | Pesticides and other micro-organic contaminants in freshwater sedimentary environments--a review[END_REF].

In order to improve water quality, the design of SUDS should focus on systems that retain all rain events, even those with the lowest intensity. To intercept ordinary events without increasing the size of storage facilities, flow rate regulation may be adjusted according to the importance of the event. It may be feasible to store the first few millimetres of rain in porous materials without discharge and by emptying via infiltration and/or evapotranspiration. As an example, "rain garden" and "bioretention" type systems are currently being promoted in other countries [START_REF] Dietz | A field evaluation of rain garden flow and pollutant treatment[END_REF][START_REF] Jefferies | Assessing the performance of SUDS for the management and control of surface water runoff in Scotland[END_REF]. These recommendations are consistent with [START_REF] Petrucci | Do stormwater source control policies deliver the right hydrologic outcomes[END_REF]. They showed that flow-rate based regulations can produce negative impacts on water discharge at the catchment scale and that volume-based regulations should be encouraged for example local infiltration facilities.

Conclusion

The research presented herein has demonstrated that the use of SUDS systems, initially aimed at peak flow attenuation, can also serve to slow and delay water flow for frequent rain events, and result in a significant reduction of the annual discharged volumes. The masses of discharged contaminants are also decreased. This phenomenon is correlated with hydraulic effects: greater initial and continuous losses limit contaminant transfer downstream while an extended residence time enhances the phenomena of substance sedimentation and adsorption. This reduction in discharged contaminants is primarily explained by a drop in runoff water volumes. These results do not systematically reveal any kind of "purifying effect" in the classical meaning (i.e. lower concentrations), but instead an overall effect of reducing mass discharges.

The effects are however highly variable from one site to another, and from one contaminant to another. They depend on the hydraulic interception of small rainfalls. For impervious storage structures the retention time of small events depends mainly on the characteristics of the flow regulator device, whereas for pervious and vegetated storage structures natural losses (infiltration, evapotranspiration) greatly contribute to the interception of frequent rain events. [START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF] have exposed the benefit of managing stormwater upstream, for the purpose of locally discharging the slightly contaminated water; refraining from mixing with heavily polluted water, and avoiding the network cross contamination process.

Nowadays, in France, the design of SUDS is mainly intended to protect against flooding and to limit discharged flows mainly by intercepting the exceptional rain events. To ensure efficiency in terms of pollutant mitigation, this upstream stormwater management approach must limit water transfer downstream and take into account frequent rain events when designed. A regulation system needs to be introduced for retaining ordinary events without excessively increasing the storage volume for ten-year return period rainfalls. As an example, the first few millimetres of rainfall could be systematically stored in a porous material or a vegetated zone without any discharge to the network but with drainage provided by infiltration and/or evapotranspiration.

  zinc r S = and zinc r S ≠ are the roof surfaces with zinc and without and i corrosionM, the mass from corrosion[START_REF] Gromaire | Zn and Pb emissions from roofing materials -modelling and mass balance attempt at the scale of a small urban catchment[END_REF].

  Downstream of the Park catchment, the majority of simulated mass values exceeded the measured values (except for SS and PAHs in the total fraction), thus indicating a mass drop due to the use of open space storage. As mentioned for the North catchment, this decreasing effect depended on the type of substance under consideration:

Fig. 1 .Fig. 3 .Fig. 4 .Fig. 7 .

 1347 Fig. 1. Methodology for comparison between sites (with H the rainfall depth, V the runoff

  CR , the runoff coefficients calculated from the volumes simulated section 2.4.1 for street, roof and garden above underground parking; p S , s S , g S the surfaces of path, street and garden; and i V the water volume measured at Reference catchment outlet.

	i CR , , s	i CR , , r	g	i
				i C , , Ref	i C , , i atm s C , ,

i build C , the concentrations from Reference catchment outlet, atmospheric deposit, street runoff and building runoff according to

[START_REF] Bressy | Towards the determination of an optimal scale for stormwater quality management: Micropollutants in a small residential catchment[END_REF]

;

Table 1 .

 1 Description of the studied catchments Roof without zinc; R=Zn: roof made with zinc; S: street; P: walking paths; Gs: garden above underground parking; G: garden

	Name	Size (ha)	R≠Zn* R=Zn*	Land use (%) S* P*	Gs*	G*	Retention system	Flow regulation
	Reference	0.82	36	7	28	3	25	0	Conventional stormwater system	-
									Vegetated roof + retention in an
	North	1.5	47	2	24	4	18	6	oversized pipe (return period of up to 2 years), possible overflow	16 l/s vortex flow regulator
									into a swale and a parking lot
	Park	2.0	12	4	0	19	26	39	2 grassed retention basins in a public park	23 l/s float valve flow regulator
	South	0.92	28	10	8	19	17	17	Swales + grassed retention basin in a square + underground tank	2 × 1 l/s nozzles + 5.6 l/s vortex flow regulator + 3 l/s pump
	Built parcel	0.13	22	43	0	4	31	0	Conventional stormwater system	-
	Street	0.031			100				Conventional stormwater system	-
	*R≠Zn:

Table 2 .

 2 Characteristics of sampled rain events (n is the number of rain events and h the cumulated water depth in mm)

		SS and organic matter	Organic micropollutants	Trace metals
	Catchment	n	h (mm)	n	h (mm)	n	h (mm)
	Reference	8	57	9	81	5	36
	North	7	65	5	58	9	99
	Park	8	73	6	66	8	94
	South	6	49	4	42	7	63

Table 3 .

 3 Runoff modelling parameters

	Surface	IL (mm) K inf (mm/h)
	Roof	0.33
	Street	0.65	0.2
	Private garden	27.6	19.7

Table 4 .

 4 Validation of micropollutant load simulations on the Reference site (t for total fraction and d for dissolved fraction)

		Measured	Simulated		Measured	Simulated
	SS (kg)	15.2 ± 1.5	15.4 ± 1.6	NPt (mg)	204 ± 18	202 ± 19
	TOC (g)	2.08 ± 0.21 2.42 ± 0.28	NPd (mg)	142 ± 15	141 ± 13
	DOC (g)	1.16 ± 0.12 1.38 ± 0.15	OPt (mg)	12.2 ± 1.3	11.7 ± 1.1
	PCBt (µg)	994 ± 99	992 ± 79	OPd (mg)	8.51 ± 1.1	8.23 ± 0.83
	PCBd (µg)	615 ± 92	695 ± 55	Cu (g)	4.50 ± 0.51 4.20 ± 0.41
	PAHt (mg)	467 ± 42	434 ± 57	Zn (g)	146 ± 16	150 ± 15
	PAHd (mg)	36.6 ± 2.9	34.9 ± 3.7	value ± uncertainty (80% confidence level)

Table 5 .

 5 Proportion of impervious surfaces and annual runoff coefficients (measured and simulated) (± measurement uncertainty) Simulated for a conventional stormwater pipe system as described in paragraph 2.4.1 TableS.1: log K OW , analytical detection limits in both dissolved and particulate fraction

	Reference	North	Park	South

*

Fluorene, phenanthrene, anthracene, fluoranthene, pyrene, benzo[a]anthracene, chrysene, benzo[b]fluoranthene, benzo[k]fluoranthene, benzo[a]pyrene, indeno[1,2,3]pyrene, di-benzo[a,h]anthracene, and benzo[g,h,i]perylene.

PCB 28, PCB

52, PCB 101 PCB 118, PCB 138, PCB 153, and PCB 180.

The box represents the 25 th and 75 th percentiles and the band inside represents the median. The ends of the whiskers depict the lowest data remaining within the 1.5 interquartile range (IQR) of the lower quartile and the highest data remaining within 1.5 IQR of the upper quartile.

Acknowledgments

This study has been conducted within the framework of the OPUR and SISTEO research programmes. The authors gratefully acknowledge the Paris Regional Council, the Seine-Saint-Denis Departmental Council, the Val-de-Marne Departmental Council, the Seine-

Annexe A.1: uncertainties calculation

Relative uncertainty (u r ) calculations of simulated or measured data were made according to the law of propagation of uncertainties.

In the following part: S is the surface, V is the volume, H is the rainfall, C is the concentration, CR is the runoff coefficient. The subscript i depends on the i th event and the subscript j depends on the type of surface: p for pathways, s for street and b for building.

For measured rainfall H i :

The uncertainty of measured rainfall is due to uncertainty of the pluviometer. V runnel is the volume of the runnel which is equal to 0.2 mm and u r (V runnel ) was estimated at 5.6 %. The relative uncertainty on the surface of the runnel, u r (S runnel ), was estimated at 1 % by the constructor. N is the number of tipping of the runnel and u r (N runnel ) was estimated at one over the whole rain.

For measured surface S:

The uncertainty of measured surface is due to uncertainty of the delimitation of the surface. The uncertainty on the delimitation of the boundaries was estimated to 0.5 m.

For simulated annual runoff coefficient (Table 5):

[ ]

was assessed from the comparison between measured data and simulated data on Reference catchment. It was evaluated at 23 % if H i > 4 mm and 47 % if H i < 4 mm.

For simulated volumes (Figure 5):

For simulated concentrations (Figure 5):

is the relative uncertainty for the concentration of the event i and the type of surface j, evaluated according to Bertrand-Krajewski et al. (2001)


M Denoual

D Brouard

A Veith

M Pouliquen

P Attia

G Allègre

Noise Study of a Heat Balanced Sigma-Delta Bolometer

Keywords: sigma-delta, bolometer, electrical substitution I

The paper presents the modeling of a heat balanced sigma-delta bolometer with Matlab Simulink®. In particular, the noise sources are modeled to enable the evaluation of the output measurement resolution.

INTRODUCTION

Bolometers are one type of infrared detector part of the thermal detectors category. A bolometer, illustrated in Fig. 1, converts absorbed IR radiation into temperature variation (P/T), which in turn changes the resistance of sensitive layer in the case of a resistive bolometer. The performance of the resistive bolometer is characterized by figures of merit such as its responsivity ( ℜ ) and its effective time constant τ eff [START_REF] Kruse | Uncooled Thermal Imaging: Arrays, Systems and Applications[END_REF]. Responsivity characterizes the voltage output variation induced by an infrared optical input. A tradeoff between low time constant τ eff and high responsivity ℜ arises from the proportional dependence according to the thermal insulation of those two figures of merit. A bolometer with high thermal insulation exhibits a high responsivity but at the cost of a high time constant.

Another limitation of existing bolometer based infrared imaging systems is due to discrepancies between the resistances of the microbolometers during the fabrication process. Discrepancies up to 10% between the resistances of the sensing resistors of an infrared imaging system can occur. This phenomenon is known as spatial noise and can represent up to 10 times the resistance variation due to the measured optical infrared signal. Such spatial noise has to be compensated by calibration and image post-processing. One opportunity exists for the reduction of the time constant of an uncooled resistive bolometer and to overcome the process discrepancies; it consists in the closed-loop mode operation of the bolometer. As for every system, the closed-loop operation of bolometers enables several performance improvements [START_REF] Denoual | Chapter 3: "Operating Uncooled Resistive Bolometers in a Closed-Loop Mode[END_REF] among which the reduction of the time constant and the operation with performance relatively independent of the performance of the sensor, i.e. here independent of the value of the resistance of the sensing resistor. Moreover, advantage can be taken from the feedback configuration to implement smart functions such as self-test, measurement range selection and working point selection [START_REF] Denoual | Chapter 8: "Smart Bolometer: Toward Monolithic Bolometer with Smart Functions[END_REF]. Operating a bolometer in closedloop assumes that one uses heat feedback means. Such association results into a heat balanced bolometer. Heat feedback means implemented in this paper is Joule power through a digital implementation of the Capacitively Coupled Electrical Substitution [START_REF] Denoual | Digital implementation of the capacitively coupled electrical substitution for resistive bolometers[END_REF]. Previous implementation used microcontroller and Pulse-Width Modulated (PWM) signals [START_REF] Denoual | Digital implementation of the capacitively coupled electrical substitution for resistive bolometers[END_REF], here a sigma-delta modulator is considered for the generation of the digital feedback signal [START_REF] Denoual | Design of a Sigma-Delta Interface for Heat Balanced Bolometer[END_REF].

Noise study is important for the adequate design of the sigma-delta modulator. Indeed on one hand, the signal-to-noise ratio of the heat balanced sigma-delta shall be limited only by the sensor and not by the conditioning or by the feedback electronics; and on the other hand, over sizing the sigma-delta interface results in higher power consumption, higher silicon surface and potentially higher design complexity. Therefore, this paper presents a model for the noise study of the heat balanced bolometer with a sigma-delta interface. The section II describes the system and the noise sources. The section III details the model used. The section IV illustrates the model through simulation results.

II. DESCRIPTION OF THE SYSTEM

A. Blocks of the System

The heat balanced bolometer system is a closed-loop system as illustrated in Fig. 1. Detailed description of the sigma-delta circuitry and of the closed-loop mode operation can be found in [START_REF] Denoual | Design of a Sigma-Delta Interface for Heat Balanced Bolometer[END_REF] and [START_REF] Denoual | Chapter 3: "Operating Uncooled Resistive Bolometers in a Closed-Loop Mode[END_REF][START_REF] Denoual | Chapter 8: "Smart Bolometer: Toward Monolithic Bolometer with Smart Functions[END_REF][START_REF] Denoual | Digital implementation of the capacitively coupled electrical substitution for resistive bolometers[END_REF] respectively. The system is composed of 6 mains parts:

• a sensor, i.e. the bolometer,

• the conditioning electronics for the amplification and the filtering, G, • a controller, C, that sets the closed-loop performance,

• a sigma-delta modulator, ΣΔ, that enables part of the analog-to-digital conversion and generates the digital feedback signal (the bitstream), • the feedback shaping block, that enables the heat feedback, • a decimator digital filter that ends the analog-to-digital conversion. The system is operating as follows. The sigma-delta modulator with the feedback shaping block constitute the feedback path of the system that compensates for the variations of the input optical power (p opt ). The controller, usually integral controller, ensures a stable temperature to the bolometer, defined by the bias voltage V Pbias , by keeping the total amount of power dissipated into the sensing resistor constant (p total =p opt +p J ). The Joule power constitutes a direct power reading of the incoming optical power and the decimator filter after the bitstream generates a digital N-bit word. The role of the feedback shaping block is to separate the thermal and electrical bias points. The sigma-delta modulator considered in this paper is the fully-differential, switched-capacitor, 2 nd order, single-loop, 1-bit DAC, sigma-delta modulator detailed in [START_REF] Denoual | Design of a Sigma-Delta Interface for Heat Balanced Bolometer[END_REF].

B. Noise Sources of the Systems

The studied noise sources are the ones from the sensor, the noise from conditioning and controller electronics, and the ones of the sigma-delta interface.

1) Noise sources of the sensor

The sensor both exhibits electrical and thermal noises. The electrical noise source of the sensor depends on the nature of the material used for the sensing resistor.

Metallic resistors mainly exhibit Johnson noise while semiconductor resistors also exhibit additional flicker noise.

The Johnson noise is expressed by

B J kTR 4 S = ( 1 )
where R B is the resistance of the sensing resistor, k is the Boltzmann constant and T the operating temperature.

The Flicker noise or 1/f noise is expressed by

( ) f K R I S f / 1 2 B B f / 1 ⋅ = ( 2
)
where I B is the bias current of the sensing resistor, K 1/f the flicker noise coefficient which value depends on the material used, and f the electrical frequency.

The thermal fluctuation noise is expressed by [START_REF] Kruse | Uncooled Thermal Imaging: Arrays, Systems and Applications[END_REF] η

ℜ = 2 eff 2 th G kT 4 S ( 3
)
where ℜ is the responsivity of the sensor, G eff is the effective thermal conductance between the bolometer and its surrounding environment [START_REF] Kruse | Uncooled Thermal Imaging: Arrays, Systems and Applications[END_REF], and η is the optical absorption coefficient of the sensor surface.

2) Conditioning and controller electronics

Conventional input referred noise power spectrum densities are used for those blocks. The noise of those blocks is characterized by Johnson and flicker noise respectively expressed by

1 n S and f / A S 1 f / 1 , 1 n = .
3

) Sigma-delta modulator

The main noise sources of the sigma-delta modulator are: the quantization noise, the noise from the front-end amplifier, the kT/C thermal noise and the noise due to clock jitter.

a) Quantization noise

The quantization noise depends on the structure of the sigma-delta converter [START_REF] Yao | Low-Power Low-Voltage Sigma-Delta Modulators in Nanometer CMOS[END_REF], in particular its order, n (the number of integrators), and on the oversampling ratio, OSR (ratio between the sampling frequency and the desired bandwidth). It is expressed according to

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + π = + ΣΔ 1 n 2 2 q OSR ) 1 n 2 ( 3 1 4 q S ( 4
)
where q is the quantum step of the digital-to-analog converter of the sigma-delta converter.

b) Fully differential switched capacitor folded cascode

In the studied structure the front-end amplifier is a fully differential folded-cascode amplifier corresponding to the design of [START_REF] Denoual | Design of a Sigma-Delta Interface for Heat Balanced Bolometer[END_REF]. As every amplifier, it exhibits both Johnson noise and flicker noise. Those are expressed respectively by

2 n S and f / A S 2 f / 1 , 2 n = .

c) Switched capacitor noise

This noise corresponds to the kT/C noise generated by thermal noise sampling at switches of the switched capacitor circuitry and is expressed by

s s C / kT C f kT 2 S ⋅ = ( 5
)
where f s is the sampling frequency and C s the sampling capacitance.

d) Clock jitter noise

Sampling clock jitter results in non-uniform sampling. The clock jitter introduces a noise due to sampling instant uncertainty that also depends on the variations of the sampled signal [START_REF] Malcovati | Behavioral modeling of switched-capacitor sigma-delta modulators[END_REF]. The sampling error is expressed by

( ) s 2 in jitter f 2 A f 2 S ⋅ τ Δ π = ( 6 )
where f in is the frequency of the input signal, A the amplitude of the sampled signal, Δτ is the standard deviation of the time sampling uncertainty assumed Gaussian [START_REF] Malcovati | Behavioral modeling of switched-capacitor sigma-delta modulators[END_REF].

III. MODELING OF THE SYSTEM

The model is realized with Matlab-Simulink® software. The white noises are modeled with white noise generators while flicker noises are obtained by filtering of white noise.

A. Models of noisy parts

The noise of the sensor is modeled by a diagram shown in Fig. 3. where both the thermal noise and the Johnson noises are obtained with white noise generators. The flicker noise is obtained through filtering of white noise. Same types of blocks are used for the noise of amplifiers (Fig. 4.) The noise of the sigma-delta circuitry is modeled with the diagram shown in Fig. 5. The Correlated Double Sampling technique (CDS) implemented in the switched capacitor sigmadelta circuitry is taken into account according to its principle, i.e. the subtraction of two consecutive noise samples [8]. Such implementation drastically reduces the offset and flicker noise effects on the total noise.

B. Entire System model

The integrators of the sigma-delta modulator are modeled according to [START_REF] Malcovati | Behavioral modeling of switched-capacitor sigma-delta modulators[END_REF] as illustrated in Fig. 6 with closed-loop unit delay cells.

The complete model is illustrated in Fig. 7. and takes into account the saturation of the electronics stages. The noise power spectrum densities of the system are obtained from temporal simulation with Matlab® Pwelch.

IV. SIMULATION RESULTS

The results section illustrates the simulation performed with the model presented. The parameters used for the sigma-delta circuitry correspond to parameters of [START_REF] Denoual | Design of a Sigma-Delta Interface for Heat Balanced Bolometer[END_REF] while the parameters used for the bolometer are typical parameters of microbolometers [START_REF] Fièque | Uncooled microbolometer detector : Recent developments at Ulis[END_REF]. Fig. 8 illustrates the effect of correlated double sampling (CDS) of the flicker noise of the front-end amplifier of the sigma-delta modulator. The noise power spectrum density is shaped by the correlated double sampling and noise at low frequency is drastically reduced. Measures performed on the sigma-delta circuitry realized [START_REF] Denoual | Design of a Sigma-Delta Interface for Heat Balanced Bolometer[END_REF] validate this behavior. Fig. 9 illustrates the output power spectrum density of the complete heat balanced bolometer as well as power spectrum density taking into account partial contributions of noise. A reference signal with 1 µW amplitude at 1 kHz is applied at the input of the system during simulation. The sampling frequency is 256 kHz for an OSR of 256 for the measure bandwidth of 1 kHz. The main shape of the power spectrum density is due to the noise shaping effect of the sigma-delta modulator that rejects the quantization noise to high frequencies. The noise of the sigma-delta modulator at low frequencies is dominated by the kT/C contribution. However this contribution is still by design lower than the contribution of the sensor that determines the noise level at low frequencies for the complete system.

V. CONCLUSION

A noise model for simulation of a heat balanced bolometer is presented. The noises sources from the sensor and the sigmadelta circuitry are taken into account. Such model is a tool for simulation that enables to determine the impact of the noise sources, of the structure of the sigma-delta modulator and of the oversampling ratio on the bit effective resolution of the circuit. Measures performed on the sigma-delta circuitry validate the models. Future work will involve measure on the complete system and cross-match with the simulation results.

Fig. 1 .

 1 Fig. 1. Top view of a bolometer pixel and equivalent electrical model.

Fig. 2 .

 2 Fig. 2. Schematic of the system.

Fig. 5 .

 5 Fig. 5. Noise models associated to the sigma-delta circuitry.

Fig. 7 .

 7 Fig. 7. Diagram of the system.

Fig. 3 .

 3 Fig. 3. Noise models associated to the bolometer.

Fig. 4 .

 4 Fig. 4. Noise models associated to the amplifiers.

Fig. 6 .

 6 Fig. 6. Schematic of the 2 nd order sigma-delta modulator.

Fig. 9 .

 9 Fig. 9. Power spectral density at the output of the heat balanced bolometer (w/-with ; w/o without).


Matthieu Denoual

Mathieu Pouliquen

Gilles Allègre

D Brouard

A Veith

Olivier De Sagazan

Patrick Attia

la diffusion de documents scientifiques de niveau

recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


M Denoual

K Fodil

C Dolabdjian

F Kauffmann

Magnetic sensor noise impact on the evaluation of magnetorelaxometry experiments

Keywords:

A complete statistical model taking into account the nature of the magnetic sensor noise is proposed for parameter extraction in magnetorelaxometry experiments on magnetic nanoparticles (MNPs). Apart from the noise, the important factors are the type of the magnetic sensor used as well as the regression model depending on the relaxation mechanism. Whatever the relaxation mechanism is, the models are nonlinear. Consequently, the relation between the signal to noise ratio of the measurement and the uncertainty of the extracted parameters is not straightforward. From the model, we derive a figure of merit that enables an a priori performance extrapolation from the characteristics of the magnetometer.

INTRODUCTION

Magnetic nanoparticles are in use for new applications in the biomedical field. They enable promising developments including manipulation, treatment and detection [START_REF] Parton | Biomedical applications using magnetic nanoparticles[END_REF]. They can be used as labels in detection techniques such as ACsusceptibility and magnetorelaxometry experiments [START_REF] Ludwig | Magnetorelaxometry of magnetic nanoparticles with fluxgate magnetometers for the analysis of biological targets[END_REF].

The objective of this work is to evaluate the impact of the nature of the magnetic sensor noise, used during magnetorelaxometry experiments, on the extraction of fitting parameters. The confidence intervals of the parameters extracted from magnetorelaxometry measurements will be used as a figure of merit to allow an a priori comparison of the performance of magnetometers taking into account their intrinsic noise as well as their dimensional characteristics. Two particular magnetometers, the fluxgate and the iGMR (improved Giant MagnetoResistance [START_REF] Perez | Advance in magnetoresistance magnetometer performances applied in eddy current sensor arrays[END_REF]) magnetometers, were chosen to illustrate the comparison scheme. Those two magnetometers were preferred because of their different nature of noise (white noise and 1/f noise behavior for fluxgate and iGMR, respectively) and their working room temperature. While fluxgates or SQUIDs have been successfully used for several years for magnetorelaxometry experiments [START_REF] Ludwig | Magnetorelaxometry of magnetic nanoparticles with fluxgate magnetometers for the analysis of biological targets[END_REF], iGMR have been more recently introduced in such measurements. In previous work, iGMR were used for magnetorelaxometry experiments for freeze-dried nanoparticles samples [START_REF] Denoual | Magnetorelaxometry using Improved Giant MagnetoResistance Magnetometer[END_REF]. The potential benefits of iGMR magnetometer used for magnetorelaxometry experiments were discussed.

The paper is organized as follows. Firstly, the potential of magnetometers for the magnetorelaxometry experiments is discussed according to their noise characteristics and respective dimensions and bandwidth. A general statistical model for the measured signal taking into account the relaxation mechanism model and noise is presented. From this model, confidence intervals of extracted parameters, used as a figure of merit, are derived in the general case from a multiplicative factor and from the standard deviation of the particular case where the magnetometer exhibiting a white noise. Abacuses are given for the estimation of the confidence interval in the general case.

II. POTENTIAL OF MAGNETOMETERS FOR

MAGNETORELAXOMETRY EXPERIMENTS

The magnetorelaxometry experiment implies the measurement of the relaxation of magnetic nanoparticles in a sample using a magnetometer in the vicinity of the sample after switching off a magnetizing field. The parameters of interest are extracted from the recorded relaxation curves through fits to non-linear models of the relaxation mechanisms. Therefore, it is obvious that the accuracy of the extracted parameters will depend on the signal-to-noise ratio of the measurement, on the fitting procedure and on the model used. This accuracy can be quantified by the evaluation of a confidence interval associated to the parameters of the given model and signal-to-noise ratio. Meanwhile, the confidence intervals are not straightforward. This paper aims at defining this relation.

Considering the magnetorelaxometry experiments, whatever the relaxation mechanism is, either Néel or Brownian, the measured signal can be expressed by

0 meas m ampl B ( t ) B B f ( t, ) n( t ) κ τ ⎡ ⎤ = ⋅ + ⋅ + ⎣ ⎦ (1)
where n(t) is the equivalent magnetic noise of the magnetometer, f(t,τ) the function expressing the relaxation mechanism, κ m a coefficient reflecting the gain factor due to the distance between the magnetometer and the sample, and B 0 , B ampl and τ are the main parameter to be extracted from the measurement for the characterization of the studied sample.

The parameters κ m and n(t) depend on the magnetometer used.

The proximity effect between the magnetometers and the sample, which depends on the magnetometer size, affects the amplitude of the measured signal, so κ m , directly. The closer to the sample the magnetometer is, the higher the measured signal is. This is true as far as the magnetometer does not disturb the measurement (e.g., via the feedback circuit or by its own magnetic material). The second term, the noise signal, depends only on the magnetometer and negatively affects the parameter extraction. Depending on the magnetometer, the noise exhibits a 1/f contribution (flicker noise) or not. From a general point of view, the higher the noise signal is, the more difficult the parameter extraction is. Thus, the poorer the confidence interval of the extracted parameter will be. But the nature of noise (white noise with or without 1/f noise contribution) shall be taken into account to elaborate the comparison of the figure of merit. Hence, the key point is to determine the relative positive and negative impact of those magnetometer dependent quantities so as to evaluate the potential of magnetometers for magnetorelaxometry experiments.

III. STATISTICAL MODEL

Here, we introduce a figure of merit that takes into account the noise contributions of the used magnetometer, especially the 1/f noise contribution. Notice that the hypothesis do not take into account any offset drift during the observation duration. This figure of merit is a multiplicative coefficient that can be used to derive the confidence interval in the general case from the confidence interval in the particular case (the magnetometer has a white noise behavior). Confidence intervals in the latter case have been previously proposed in [START_REF] Denoual | Magnetorelaxometry using Improved Giant MagnetoResistance Magnetometer[END_REF].

A. Noise model

In order to estimate and to obtain statistical properties of estimated parameters B ampl and B 0 , we propose to model noise of magnetometer, n(t), with a parametric family of stationary Gaussian noise n α,d,σ up to second order depending on three parameters α ∈ [0, 1], d ∈ [0, ½] and σ ∈ [0, +∞], whose power spectral density with the normalized frequency

f ∈ [-1/2, 1/2], is defined by ( ) ( ) 2 2 2 2 2 1 1 1 d n ,d, d i f / K S f e α σ π σ α α - ⎛ ⎞ ⎜ ⎟ = - + ⎜ ⎟ - ⎝ ⎠ (2) with 1 2 2 1 2 2 1 1 / d d / i f K df e π - - = - ∫ .
In the time domain, we can find two independent white Gaussian noise contributions of unit variances ε 1 , ε 2 such that

( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + - = - - 2 1 1 2 , , 1 1 ε α ε α σ σ α d d d z K n (3)
where z -1 is a delay operator, and

(1-z -1 ) -d is a fractional integrated operator, (1-z -1 ) -d ε 2 is in a AutoRegressive
Fractionally Integrated Moving Average process named ARFIMA(0,d,0) [START_REF] Brockwell | Times series: theory and methods[END_REF][START_REF] Taniguchi | Asymptotic theory of statistical inference for time series[END_REF][START_REF] Beran | Statistics for Long-Memory Processes[END_REF][START_REF] Hosking | Fractional differencing[END_REF].

The interpretations of these three parameters are the following:

σ 2 is the variance of n(t), as the power spectral density is a decreasing function of f in the interval [0, ½], it means that the level of power density of white noise of n α,d,σ can be defined by its value for f = ½ denoted ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = d d d K W 2 2 2 2 , , 2 1 α α σ σ α .
Therefore (1-α 2 ) can be interpreted as an approximation of the percentage of the power of white noise which is included in the total power of the power spectral density. When α = 0, one finds that n α=0,d,σ = σ ε 1 and Sn α=0,d,σ (f) = σ 2 which means n α=0,d,σ (t) is a white Gaussian noise of variance, σ 2 , without any 1/f contribution at f = 0 + . When α = 1, n α=1,d,σ is a purely autogressive, fractionary integrated and moving average. This noise models the noise contribution of magnetometers exhibiting a 1/f noise behavior.

In fact, if α > 0 then n α,d,σ is considered as a long memory process because ( )

+ → ≈ 0 , / ) 2 ( 2 2 2 2 , , f f K f S d d d d n α π σ σ α or ( ) ( ) ( ) 2 0 n ,d, log S f d log f , f α σ + ≈ - → .
Therefore, the parameter 2d represents the slope of power function in a log-log scale. The frequency corner, ( )

d d d corner W K f 2 / 1 , , 2 2 / 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = σ α α σ π
, of a noise n α,d,σ is the frequency for which the asymptote (f→0 + ) meets the noise density baseline S=W α,d,σ , as illustrated in Fig. 1 in the case W α,d,σ =1.

( ) ( ) ( ) ( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Γ + - Γ - Γ + Γ + - = d d h d h d h h h d n 1 1 ) ( 2 0 2 2 , , α δ α σ γ σ α (4)
where δ 0 is the Dirac function and Γ is the gamma function.

B. Regression model

To estimate the unknown parameters, we propose to use an Expectation-Maximization (EM) type algorithm, which can be described with the following steps:

• searching an admissible solution, and whereas the parameters are not converging, • maximize the likelihood of the deterministic model knowing the stochastic error model,

• maximize the likelihood of the stochastic error model knowing the deterministic model.

Let t i (= t delay + i/F e ) 0≤i≤N be the observation times in second, where F e is the sampling frequency and t delay is the delay between the beginning of the relaxation and the beginning of the acquisition, then the measured signal can be modeled by ( ) ( ) ( )

N i t n t f B B t B i d i ampl m m i meas ≤ ≤ + + = 0 , , , , 0 σ α τ κ κ (5) Let ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = τ , 1 ) , ( i t f t i z
, then the model can be written for

0 ≤ i ≤ N as ( ) ( ) meas i ,d , i B t z(i, )' n t α σ τ β = + (6)
where

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 0 β β β
is the column vector of the two unknown parameters with β 0 representing κ m B 0 and β 1 representing κ m B ampl . τ is a vector of parameters of the relaxation model which is supposed to be known. As the model of noise covariance matrix is parametric, depending on the parameters, α, d, σ, which are assumed to be known, the estimation of the unknown parameters β 0 and β 1 can be derived using the weighted least square method (best linear unbiased estimator).

Classically, let N be the number of observed samples and

( ) ( ) ' , d , , d , N ) N ( n , ), 0 ( n cov σ α σ α = Σ …
the covariance matrix of the first N+1 samples of the noise model ( )

0 ,d , ,d , n ( ), ,n (N ) α σ α σ … is given by ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = Σ ) 0 ( ) ( ) ( ) 0 ( n n n n N N N γ γ γ γ . ( 7
)
Let Y N be the first observed N+1 samples of a relaxation signal. The design matrix Z N of the regression model can be written as

( ) ( ) ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = α α , 1 , 1 0 N N t f t f Z . ( 8
)
Then, if the rank of Z N = 2, the Best Linear Unbiased Estimator (BLUE) is uniquely defined by ( )

1 1 1 BLUE ' ' N N N N N N ˆZ Z Z Y β - - - = Σ Σ . ( 9
)
With the hypothesis that the physical parameters τ and parameters α, d, σ of the noise model are known, the estimated parameters BLUE β is a Gaussian vector with mean β (the true one) and covariance matrix

( ) ( ) ( ) ( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Σ = - - 1 0 1 1 0 0 1 1 ' BLUE var , ĉov , ĉov var ĉov β β β β β β β N N N Z Z . (

C. Figure of merit

As a figure of merit of a magnetometer, we propose the following criteria Merit( ,d,,f ,,F ,N ,

where the standard deviation σ(α, d) is chosen such that the power density at the highest frequency is one.

This figure of merit of magnetometer can be interpreted as a merit ratio between this magnetometer and the same magnetometer with no 1/f contribution (α = 0). It expresses for the impact of 1/f noise contribution on the precision of the estimation of the unknown parameters is x times worse compared to the precision of estimation in the case of a magnetometer having the same white noise level. A magnetometer exhibiting only white noise has a figure of merit of 1, in particular, its figure of merit does not depend on the relaxation function, whereas the figure of merit for magnetometer exhibiting a 1/f noise contribution depends on the relaxation function

) , , , , , 1 , , 0 ( ) , , , , ), , ( , , ( ) , , , , , , , ( delay e

D. Experimental performance comparison

Fluxgate and iGMR magnetometers are used as examples. The relaxation model for liquid samples (mobile magnetic nanoparticles) is taken as

3 1 2 t f ( t, ) exp τ τ τ τ ⎛ ⎞ ⎛ ⎞ - ⎜ ⎟ = -⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ . ( 14
)
with τ 1 = 0 s, τ 2 = 1.28 µs, τ 3 = 0.23. We simulate N = 2048 samples with a sampling frequency F e ≈ 100 or 200 kHz and a null delay t delay = 0. These values correspond to experimental data.

For the relaxation of freeze-dried samples (immobilized magnetic nanoparticles) the following model is given by

1 1 f ( t, ) ln t τ τ τ ⎛ ⎞ = + ⎜ ⎟ - ⎝ ⎠ . ( 1 5 )
with τ 1 = 0 s and τ 2 = 2 s. We simulate N = 4096 samples with a sampling frequency F e = 5 kHz and a delay t delay = 1/F e . These values correspond to the experimentally realized data presented in previous work [START_REF] Denoual | Magnetorelaxometry using Improved Giant MagnetoResistance Magnetometer[END_REF].

As noise for the used fluxgate magnetometer is nearly white in the considered range of frequency, we can take the model for noise n α,d,σ (white noise of variance σ 2 ). For the iGMR noise, we can take the family noise n α,d,σ with a fixed As the fluxgate noise is white, the figure of merit of this magnetometer is one. For iGMR magnetometers, we can study the figure of merit in the case of liquid and freeze-dried samples. The parameter α is obtained from Fig. 2 after normalization of the corner frequency experimentally observed. For example, an iGMR used for experiments with a 200 kHz sampling frequency and a noise corner frequency of 10 kHz, would have a normalized corner frequency equals to 0.05. Knowing this value the closest curve in the abacus Fig. 2 gives a parameter α of 0.999822. The figure of merit is then obtained from Fig. 3. Fig. 3.a. corresponds to the case of liquid samples relaxing via the Brownian mechanism and studied in this paper while Fig. 3.b corresponds to the case of freeze-dried samples. For example, considering the case of liquid sample, and an iGMR magnetometer, the normalized corner frequency derived from experiments implies a 1.33 multiplication factor to derive the standard deviation compared to a magnetometer with the same white noise level and no 1/f contribution. As mentioned previously, in the case of magnetometers exhibiting a 1/f noise contribution, the figure of merit also depends on the relaxation mechanism. Especially, the simulation study indicates that the 1/f noise contribution in magnetometer noise has less impact on experiments with liquid samples than with freeze-dried samples.

IV. CONCLUSION

A model that takes into account the nature of the noise of magnetic sensor is proposed. It enables the evaluation of the confidence interval of the parameter extracted during magnetorelaxometry experiments.

Fig. 1 :

 1 Fig. 1: Normalized power spectral noise density for 1 , 6 . 0 , 1 . 0 = = = σ α d

  10) So, we can define a figure of merit of all magnetometers by the standard deviation precision which can be obtained to estimate the concentration κ m B ampl . This figure of merit is given by the characteristics of magnetometer noise α, d, σ, the geometry of the whole sample and the magnetometer κ m , the relation function f and its parameter τ and acquisition parameters F e , N, t delay .

2

 2

3 αFig. 2 :

 32 Fig. 2: Normalized power spectral noise density versus normalized frequency for 1/f noise modeling.

Fig. 3 :

 3 Fig. 3: Figure of merit depending on relaxation mechanism for magnetometer exhibiting 1/f noise contribution: case of liquid sample a), case of freeze-dried samples b).