Samer Majdalani
email: samer.majdalani@umontpellier.fr

Vincent Guinot

Solute transport in heterogeneous model porous media under different flow rates: experimental and modelling study

Keywords: flow rate, solute transport, heterogeneous model porous media, advectiondispersion, multi-region

Highlights:

-High quality experimental data set for passive transport in heterogeneous porous media -Flow rate independent and continuity preserving formulation for MRAD parameters confirmed by experiments -Linearity of exchange and dispersion and exchange coefficients with respect to flow rate confirmed by the experimental data set

Introduction

Laboratory experiments on model heterogeneous porous media MHPMs contribute to understand the behaviour of solute transport in real world situations [START_REF] Majdalani | Solute transport in periodical heterogeneous porous media: Importance of observation scale and experimental sampling[END_REF]. A large number of laboratory scale experimental studies on solute transport in porous media involve a single flow rate, while in real world situations the flow rate may vary over one or two orders of magnitude. Several laboratory studies of solute transport in porous media have tested a variable flow rate in order to test its effect on modelling parameters. A wide review can be found in [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF], [START_REF] Maraqa | Prediction of mass-transfer coefficient for solute transport in porous media[END_REF], and [START_REF] Haggerty | What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results[END_REF]. In these studies, the most widely used model is the Mobile-Immobile Model (MIM), based on the advection-dispersion equation where two regions (mobile and immobile) exchange mass between each other. The review of [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF] is based on 20 experiments. It shows correlations between the flow rate and the mass transfer coefficient, the mass transfer and the advection time scale, the mass transfer and diffusion time scales, the mass transfer and longitudinal interaction time scales. The review by [START_REF] Maraqa | Prediction of mass-transfer coefficient for solute transport in porous media[END_REF] is based on 19 experiments. It shows that correlations exist between the mass transfer coefficient and the flow rate, or the mass transfer coefficient and the residence time. The review of [START_REF] Haggerty | What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results[END_REF] is based on 316 experiments. Correlations are inferred between the effective mass transfer time and other parameters such as the flow rate, the capacity coefficient, the advective residence time and the experimental duration. Experiments reported in the literature involve porous media made of sand [START_REF] Coats | Dead-end pore volume and dispersion in porous media[END_REF][START_REF] Gaudet | Solute transfer with exchange between mobile and stagnant water through unsaturated sand[END_REF][START_REF] De Smedt | Study of tracer movement through unsaturated sand[END_REF][START_REF] Rambow | Laboratory method for studying pesticide dissipation in the vadose zone[END_REF]Lennartz, 1993, Kookana et al., 1993 ;[START_REF] Sharma | Study on non-Fickian behaviour for solute transport through porous media[END_REF][START_REF] Sutton | Characterizing the hydraulic and transport properties of a constructed coarse tailings sand aquifer[END_REF], loam (van Genuchten and Wierenga, 1977;van Genuchten et al., 1977), loamy sand [START_REF] Khan | A laboratory study of the dispersion scale effect in column outflow experiments[END_REF], clay [START_REF] Jørgensen | Modeling of non-reactive solute transport in fractured clayey till during variable flow rate and time[END_REF], stony soil [START_REF] Schulin | Solute transport through a stony soil[END_REF], glass beads [START_REF] Krupp | Miscible displacement in an unsaturated glass bead medium[END_REF][START_REF] De Smedt | Solute transfer through columns of glass beads[END_REF][START_REF] Berkowitz | Laboratory experiments on dispersive transport across interfaces: The role of flow direction[END_REF], field soil [START_REF] Smettem | Soil-water residence time and solute uptake, 3, Mass transfer under simulated winter rainfall conditions in undisturbed soil cores[END_REF][START_REF] Sutton | Characterizing the hydraulic and transport properties of a constructed coarse tailings sand aquifer[END_REF], loam and field soil [START_REF] Selim | Transport and ion exchange of calcium and magnesium in an aggregated soil[END_REF], aggregate [START_REF] Rao | Solute transport in aggregated porous media: Theoretical and experimental evaluation[END_REF][START_REF] Seyfried | Solute transport in undisturbed columns of an aggregated tropical soil: Preferential flow effects[END_REF][START_REF] Koch | Non-reactive solute transport with micropore diffusion in aggregated porous media determined by a flow interruption method[END_REF][START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF][START_REF] Nkedi-Kizza | On the equivalence of two conceptual models for describing ion exchange during transport through an aggregated oxisol[END_REF][START_REF] Brusseau | Simulating solute transport in an aggregated soil with the dual-porosity model: Measured and optimized parameter values[END_REF][START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF], or spherical clayey inclusions in sandy media [START_REF] Ngoc | Non-Fickian dispersivity investigation from numerical simulations of tracer transport in a model double-porosity medium at different saturations[END_REF]. In the aforementioned studies, porous media are either homogeneous or moderately heterogeneous. Heterogeneity is generally obtained by introducing coarse aggregates into a finer surrounding medium (e.g. glass or clay beads into sand). [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF] introduced random porous polyethylene cylinders (porosity 0.50) into silt soil (porosity 0.43). [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF] used a porous medium consisting of fine sand and polyethylene cylinders in different proportions for 3 experimental columns of different lengths, with a respective total porosity of the composite medium of 0.34, 0.36, and 0.40. [START_REF] Berkowitz | Laboratory experiments on dispersive transport across interfaces: The role of flow direction[END_REF] used a porous medium made of adjacent porous materials of fine (1 mm) and coarse (4 mm) glass beads, where the porosity (0.39) is the same in both fine and coarse media. Therefore, in literature studies, the porosity of the introduced heterogeneity mostly belongs to the range [0.5, 0.6], which is rather close to the porosity of the surrounding medium (range [0.3, 0.4]). In many laboratory studies the flow rate varies over a single order of magnitude [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF][START_REF] Maraqa | Prediction of mass-transfer coefficient for solute transport in porous media[END_REF][START_REF] Haggerty | What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results[END_REF]. Although reviews [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF] report flow rate variations over several orders of magnitude, they aggregate several studies made on different media, that were all performed over one (or slightly more) order of magnitude. Rare are the studies where the flow rate in laboratory heterogeneous porous media is varied over two orders of magnitude within the same medium. One of the widest flow rate ranges is reported in [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF], where the pore velocity is varied by a factor 54 and only 3 different flow rates are used. In the study by [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF], the pore velocity is varied by a factor 12.5. As mentioned above, the laboratory porous medium of [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF] and [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF] is moderately heterogeneous. Moreover, the flow direction in these studies is vertical upward (from bottom to top of the column). We argue that salt-based solute transport experiments under upward flow are liable to induce density effects and that the observed BreakThrough Curves (BTCs) might not reflect the real behaviour of passive transport. In fact, rare are the studies of solute transport in porous media using salt tracer that take the possibility of solute density effects into account and their impact on BTCs. Due to gravity, strongly concentrated solute may be trapped in lower parts of the porous medium, thus impacting the behaviour of solute transport and biasing the breakthrough curves compared to those of purely passive transport. Another issue concerning experimental data is reliability. In the aforementioned studies, rare are the solute transport experiments where both the experiment and the porous medium are replicated. The first point means that one should repeat the same experiment several times on a given porous medium column, while the second point means that one should construct several identical porous medium columns and that the ensemble of the experimental protocol should be repeated on each of them. By taking the mean of all replicates, as well as the minimum and maximum values, the statistical variability of the breakthrough curves stemming from various sources can be assessed. Not only is statistical variability of great help in estimating modelling parameters, but it also gives stable and reliable breakthrough curves by minimizing realization sampling effects. In this paper, we study solute transport in laboratory scale Model Heterogeneous Porous Media (MHPMs) under different flow rates. The flow rate Q of tracer step experiments varies over two orders of magnitudes: the ratio of the maximum to the minimum flow rate is 100. The flow rate is modified across the different realizations/experiments but not within a single realization (the flow in each given experiment is stationary). The Model Heterogeneous Pours Medium (MHPM) used in the present study is strongly heterogeneous. A rectangular cavity (porosity 100%) is introduced into surrounding glass beads (porosity 40%). The purpose is to assess the applicability of standard transport models to strongly heterogeneous media, in order to determine whether the degree of heterogeneity induces limitations in the model. Besides, solute transport experiments are done with horizontal flow and we make sure that no solute density effect is present in the MHMP. All tracer step experiments are replicated (see Section 2): each experiment is replicated 3 times and the ensemble of the experimental protocol is replicated on 3 different MHPMs built under the same specifications. We simulated the breakthrough curves by a classical approach based on the multi-region advection-dispersion (MRAD) model as presented and analyzed in [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF], where only two mobile regions are used. The MRAD model is a generalization of well-known models such as the Two Region (TR), Mobile-Immobile (MI) and Multi-rate models [START_REF] Coats | Dead-end pore volume and dispersion in porous media[END_REF][START_REF] Griffioen | Interpretation of two-region model parameters[END_REF][START_REF] Haggerty | On the late-time behaviour of tracer test breakthrough curves[END_REF][START_REF] Van Genuchten | Mass Transfer Studies in Sorbing Porous Media I[END_REF]. A number of studies have focused on how the model parameters behave with the flow rates [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF][START_REF] Haggerty | What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results[END_REF][START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF]. They do not all lead to the same conclusions and a wide variety of behaviours is reported. Little is known about how the parameters were constrained in the model calibration process. In particular, it is not always clear whether water and solute mass conservation were enforced in the calibration process. In other words, it is not always clear whether the flow rate in the model is identical to that prescribed in the experiments. Not preserving this condition may yield biases in the calibrated transport parameters. In the present work, mass conservation is enforced by rewriting the MRAD model in terms of flow rate-independent parameters, among which a discharge partition coefficient for the various regions in the model. To our best knowledge, such a reformulation has not been presented in the literature before. Obviously, many other modelling formalisms than the MRAD might be used. More complex models such as Multirate models [START_REF] Haggerty | Multiple-rate mass transfer for modelling diffusion and surface reactions in media with pore-scale heterogeneity[END_REF] and fractional dynamics / Continuous-Time Random Walk (CTRW) models have proven efficient in the modelling of tailing effects and transient dispersion [START_REF] Berkowitz | Non-Fickian transport and multiple-rate mass transfer in porous media[END_REF][START_REF] Bijeljic | Pore-scale modelling and continuous time random walk analysis of dispersion in porous media[END_REF][START_REF] Le Borgne | Effective porescale dispersion upscaling with a correlated continuous time random walk approach[END_REF][START_REF] Sun | Use of a variable-index fractionalderivative model to capture transient dispersion in heterogeneous media[END_REF][START_REF] Moradi | Modeling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection-dispersion equation[END_REF]. Multi-region versions of these approaches have been presented recently [START_REF] Sun | An efficient approximation of non-Fickian transport using a time-fractional transient storage model[END_REF][START_REF] Yin | A dual heterogeneous domain model for upscaling anomalous transport with multi-peaks in heterogeneous aquifers[END_REF]. Such models are particularly efficient in modelling the long-term behaviours and tailing effects in breakthrough curves that are typical of anomalous transport. They can account for asymptotically long travel times resulting from trapping as well as sudden jumps over long distances, known as Levy flights [START_REF] Klafter | Stochastic pathway to anomalous diffusion[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. However, in the present study, heavy tailing was not identified as a salient characteristic of the BTCs. For this reason, using this type of model was not deemed necessary. The present paper is organized as follows. Section 2 presents the materials and methods. Subsections 2.1 and 2.2 are devoted respectively to the experimental setup and protocol for obtaining a wide range of transport conditions in a strongly porous heterogeneous medium, with flow rates spanning two orders of magnitude. Subsection 2.3 presents a continuityenforcing writing of the MRAD model. Subsection 2.4 deals with the calibration approach, that takes experimental imprecision into account. Section 3 presents the experimental and modelling results. Sections 4 and 5 are devoted respectively to a discussion and conclusion.

Material and Methods

Model heterogeneous porous medium Xi (i = 1 to 3) and Yi (i = 1 and 2)

Three parallelepiped (Length 20 cm, height 2 cm, width 10 cm) columns (X1, X2, and X3) were built manually by pasting 5 mm thick plastic plates. The Xi columns are labelled MHPM A, MHPM B, and MHPM C hereafter. They all contain a parallelepiped pore (length 15 cm, height 2 cm, width 1 cm) on one side. The rest of the column is filled with 1 mm glass beads (Figure 1). The rectangular pore (100% porosity) plays the role of a heterogeneity where water rapidly flows in comparison to the surrounding 1 mm glass beads (40% porosity). The ends of the rectangular pore are covered with 500 µm sieve so that the 1 mm glass beads cannot penetrate into it. The Xi columns have four inlets and four outlets. One inlet/outlet faces the rectangular pore, the remaining three inlets/outlets are spread over the width of the MHPM. The pore volume of each MHPM is V0 = 158.8 ml. To reduce solute density effect, the height of the MHPMs was taken very small (2 cm) in comparison to the width (10 cm) and length (20 cm). The assumption of negligible density effects was verified as follows. Two rectangular columns Y1 (Figure 2) and Y2 (Figure 3) were designed in a manner that if they are turned upside down, the position of the heterogeneity would change along the middle horizontal plane. Thus, should solute density effects be present, solute transport differ for the Up and Down positions in both Y1 and Y2 columns, and the resulting BTCs would be different.

Tracer step experiment: setup and protocol

The solute transport is studied using tracer step experiments under saturated conditions. The columns are first filled with deionized water, then a step of salty water (NaCl) with a concentration of C0 = 0.05 M is introduced into the column via a peristaltic pump (Lead Fluid TM ). This yields an Atwood number A = 2.9×10 -3 , suggesting that density effects will be negligible. The concentration at the column output is given by a conductivity meter (WTW TetraCon 325 TM ). The flow rate is estimated by weighing the effluents on a scale (Figure 4). The conductivity meter and the scale are connected to a data logger that samples the measurements at regular time intervals. The step experiment is stopped when the concentration at the column output reaches C0 (concentration at the column input). The flow is maintained with pure, deionized water until the conductivity of the outflowing concentration returns to its initial, background level. Each of the three Xi columns underwent step experiments with seven flow rates varying from 0.25 to 25 l/h (Table 1). The flow rate is varied across different realizations/experiments but not within a single experiment (the flow is constant during a given step experiment). Since the tested flow rates vary in ratios of 2, 4, 10, 20, 40, 100 and 200, they are lalelled Q2, Q4, Q10, Q20, Q40, Q100, and Q200 hereafter. For the lowest flow rate (Q2), the duration of the step experiment was 3 hours and the data sampling interval wass 10 seconds. For the higher flow rate (Q200), the duration of the step experiment was 2 minutes and the data sampling interval was 1 second (Table 1). Each step experiment was replicated three times for each of the three Xi columns, hence a total 7 × 3 × 3 = 63 experimental BTCs (see Results and Discussion sections). The Y1 and Y2 columns underwent step experiments with the seven flow rates indicated in Table 1 but the duration of the Yi experiments are smaller than those of Xi because the volume of Yi columns is smaller than that of Xi columns. A selection of the BTCs for the Yi columns is coemmented in the Results and Discussion sections.

MRAD Model

Model presentation

The so-called Multi-Region Advection-Dispersion (MRAD) [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF]) model is a generalization of a variety of models introduced in the literature from the 1970s. Two wellknown particular configurations are (i) the Two Region (TR) model [START_REF] Coats | Dead-end pore volume and dispersion in porous media[END_REF][START_REF] Griffioen | Interpretation of two-region model parameters[END_REF]) and its particular implementation known as the Mobile-Immobile (MI) model [START_REF] Van Genuchten | Mass Transfer Studies in Sorbing Porous Media I[END_REF], and (ii) the MultiRate (MR) model [START_REF] Haggerty | On the late-time behaviour of tracer test breakthrough curves[END_REF]. The general model consists of R regions, each of which may be mobile or immobile with its own flow and dispersion characteristics, exchanging mass with each other. The governing equations can be written in vector form as (1a)

(1b) (1c) (1d) (1e)
Where ci, Di, ui, αi, and µi are respectively the concentration, dispersion coefficient, flow velocity, dispersivity and volume fraction for Region i (i = 1,…, R), and kij is a first-order exchange coefficient between Regions i and j. The antisymmetry property for the matrix K and the zero sum in Equation ( 1d) are necessary conditions for mass conservation [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF]. Specifying appropriate combinations allow arbitrary exchange functions between the mobile and immobile regions to be modelled, including transfer functions with multiple transfer time scales and long-term memory effects [START_REF] Haggerty | On the late-time behaviour of tracer test breakthrough curves[END_REF][START_REF] Haggerty | What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results[END_REF]. The model is also able to represent the transition from ballistic behaviour (i.e. variance of solute position growing proportionally to the square of travel time) at small spatial and time scales to normal (i.e. Fickian) dispersion at large scales [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF]. Moreover, Fickian behaviours can be obtained without including any dispersion terms by specifying appropriate velocities and exchange coefficients between parallel regions [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF]). In the most general possible layout (Figure 5), each region may exchange with all the other regions. Intuition suggests that exchange should be possible only between two regions that have similar flow velocities. This would ensure a smooth transition between low velocity (or immobile regions) and fast flowing regions. However, in a strongly heterogeneous porous medium, fast flowing regions may be located in the immediate neighbourhood of slow flow regions (Figure 5a), and the regions may not necessarily be connected. Therefore, as far as flow topology is concerned, nothing should preclude an exchange between any two different flow regions in the general case. The exchange coefficient in the model of Figure 5b reflects not only the transfer rate between the various regions, but also their degree of connectivity.

Identifying a flow rate-independent formulation for the MRAD model from experimental breakthrough curves

The question arises of how the model parameters, i.e. the coefficients of the matrices U, D and K vary with the flow rate Q. The simplest possible assumption is that D, K and U are proportional to the discharge and that µ is fixed. These assumptions can be checked easily from the experimental breakthrough curves. To do so, the following variable change ω = Qt/V0 is introduced, where V0 is the flow volume in the domain. This volume can be determined from the experimental breakthrough curves cB(t) by noting that the inflow concentration signal cin is a step function with amplitude c0:

(2) which yields (3)

The integral is computed from the experimental values using the trapezium rule for numerical integration. Performing the variable change ω = Qt/V0 in Equation (1a), using (1b)-(1e) leads to (4)

In the particular case where K and U are proportional to Q and the dispersivities αi are independent of Q, all the matrix coefficients in equation ( 4) are Q-independent and all plots of c (x, ω) obtained for different values of Q superimpose. From an experimental point of view, only the BTC is known:

(5)

where xds is the abscissa of the downstream end of the model. Therefore, for the BTCs cB(ω) to superimpose, the volume fraction matrix µ must be independent of Q.

New MRAD formulation

The MRAD model studied here is not different conceptually from those presented earlier in the literature. The only difference lies in the presentation of the equations and in the identification of the model parameters. The proposed formulation brings the following improvements over previously published versions:

the (assumed) proportionality of the dispersion and exchange coefficients to the flow rate is incorporated directly in the formulation, the new formulation intrinsically enforces conservation of both water and solute regardless of the flow rate. Such conservation is not enforced in usual calibration approaches (such as that described in Majdalani et al. Substituting the relationships ( 6) into ( 7) yields ( 8)

The vector form of the equation is

(9a) (9b) (9c) (9d)
where Kij and kij stand respectively for the elements of the matrix K and the exchange coefficients between the regions. Equation ( 9a) is rewritten as

(10a) (10b)
Noting that A = V0/L, (L is the length of the column) one has (11)

For a two region model as dealt with in the present work, there are 5 independent parameters: α1, α2, µ1, ν1, and k12. Note that µ2 = 1 -µ1 and ν2 = 1 -ν1. In the absence of dispersion and exchange (a1 = a2 = k12 = 0), the analytical solution for a step injection consists of two steps arriving at the downstream end of the MHPM at two different times (dashed line in Figure 6, top). Increasing the dispersion coefficients 1 and 2 tends to smooth out these steps (solid line in Figure 6, top). (µ1; µ2) act on the contrast between the arrival times of the two steps, while ν1 influences their relative sizes. Increasing k12 tends to smooth out the transition between the two steps (Figure 6, bottom). Note that the above considerations are only indicative of the broad influence of the various parameters on the modelled solution. They do not reflect the interactions between the exchange and dispersion parameters that may have similar effects on the long term behaviour of the solution. For instance, the theoretical analysis in [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF] shows that a Fickian dispersive behaviour is obtained asymptotically for 1 = 2 = 0, provided that the exchange coefficient k12 is non-zero. Note that the coefficients in matrix K have the dimension of the inverse of a length. Owing to the conservation properties (9c), (9d), K has at least one nil eigenvalue, while the remaining ones are all negative [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF]. The smaller of the absolute values of the remaining eigenvalues gives an order of magnitude of the distance needed for the concentrations in all regions to homogenize. Beyond this distance, the standard, single region, Fickian dispersion model becomes valid, even if the dispersion coefficients are zero in all regions [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF]. In the case of a two region model as explored in the present study, the non-zero eigenvalue is equal to -2k12, which yields a characteristic distance (2k12) -1 . Note that this holds in the case of zero dispersion coefficients αi in all regions. In the presence of dispersion, front spreading occurs faster in all regions and the normal, Fickian dispersion regime is achieved over smaller distances.

Calibration

The model parameters are calibrated as introduced in Majdalani et where cB and cexp are respectively the simulated and experimental breakthrough concentrations, t0 > 0 and T are respectively the simulation time step and simulation length, and δc is the measurement precision. The max() operator ensures that the modelling error is zero if the difference between the simulated and experimental concentrations is smaller than the measurement precision δc.

The square root of J thus provides a measure of the difference between the modelling results and the experiments. It is to be compared with the experimental imprecision, that is measured by computing the standard deviation of the various replicas of a given experiment: (13a)

(13b) (13c)
where is the experimental concentration measured at time t for the jth replica of the ith flow rate, mi is the number of replicas done for the ith flow rate, N = 6 or 7 is the number of flow rates tested, and Ti is the length of the experiment for the ith flow rate. is therefore the average at time t of all the concentrations obtained from the mi replicas done for the ith flow rate, and is a measure of the experimental dispersion, at a given time t, of the various replicas for the ith flow rate. Averaging over time and over the N flow rates yields av, a measure of the overall dispersion of the experimental concentration time series about the mean signal for all times and all flow rates.

Results

Solute density effect (Yi columns)

Columns Y1 and Y2 are only test columns to verify the existence or the absence of solute density effect. Since the position of the heterogeneity is not the same in Yi columns whether they are in Up or Down position (Figures 2 and3), and since the solute has a tendency to dive to the bottom due to gravity (water + solute is denser than deionized water), gravity effects would induce differences between the BTCs of Columns Y1 and Y2 would be different in Up and Down positions. We made step experiments on Columns Y1 and Y2 in Up and Down positions with all the flow rates given in Table 1 and we noticed that the BTCs were the same (in the limit of experimental uncertainty) in Up and Down positions, regardless of the flow rate. As an illustration, Figure 7 (top) (respectively bottom) shows the breakthrough curves of column Y1 (respectively Y2) in Up and Down positions for Flow rate Q100. Therefore, as argued in Section 2, solute density effects are negligible with the present MHPM design.

Flow rate effect (Xi column or MHPM A, B, and C)

Each of the three Xi columns underwent step injections under the seven flow rates Q2 -Q200, with three replicates in each case. As suggested previously in the MRAD formulation, we adopt the variable change ω = Qt/V0 to plot the BTCs. Figure 8 shows that, for every MHPM (A, B, and C), the resulting experimental breakthrough curves superimpose in the limit of experimental imprecision for all flow rates. This validates the assumption of a model formulation with discharge-independent parameters, as expressed by Eq. ( 4).

In homogeneous porous media, BTCs classically exhibit an 'S' shape. The BTCs shown in Fig. 8 have a double 'S' shape: the lower 'S' is attributed to a fast solute transport mode through the heterogeneity (rectangular pore), while the higher 'S' is attributed to a slower transport mode within the glass beads. Other studies have also noticed that BTCs in heterogeneous porous media do not have a classical 'S' shape [START_REF] Silliman | Laboratory evidence of the scale effect in dispersion of solutes in porous media[END_REF][START_REF] Golfier | Comparison of theory and experiment for solute transport in weakly heterogeneous bimodal porous media[END_REF][START_REF] Majdalani | Solute transport in periodical heterogeneous porous media: Importance of observation scale and experimental sampling[END_REF]. Double 'S' shaped BTCs (or double peak curves for their derivatives) have also been reported in natural/real media [START_REF] Maloszewski | Mathematical modelling of tracer experiments in the karst of Lurbach system[END_REF][START_REF] Goldscheider | Tracer tests in karst hydrogeology and speleology[END_REF][START_REF] Perrin | Inference of the structure of karst conduits using quantitative tracer tests and geological information: example of the Swiss Jura[END_REF][START_REF] Field | Solute transport in solution conduits exhibiting multi-peaked breakthrough curves[END_REF][START_REF] Dewaide | Double-peaked breakthrough curves as a consequence of solute transport through underground lakes: a case study of the Furfooz karst system Belgium[END_REF].

MRAD simulations results

The calibration results are given in Table 2. Noting that the error J is the product of two Root Mean Square Errors (RMSEs), one with respect to t and the other with respect to 1/t, J 1/2 gives a meaningful assessment of the difference between the simulated and experimental breakthrough curves. J 1/2 is about 3% of the total variation in the concentration, which can be deemed a very good agreement between the model and the experiment. In comparison, the experimental imprecision ranges from 0.56% to 1.56% (i.e. half of the modelling error).

Figure 9 shows the experimental versus simulated breakthrough concentrations. A satisfactory alignment over the first bisector is observed. This was expected considering the values of J 1/2 in Table 2.

Since the behaviour of the breakthrough curve for the Q200 setting in Figure 8 seems different from that of the other BTCs, a second calibration of the model is run by removing the Q200 time series from the experimental data set. This leads to slightly better calibration results (Table 3), with J 1/2 reduced by 10% to 25% compared to the calibration results in Table 2.

Removing Q200 from the experimental data set leaves the average experimental imprecision almost unchanged. Comparing the average calibrated parameter values in Tables 2 and3 shows that the calibrated 1, 1 and k12 are significantly sensitive to the presence of Q200 in the data, while a2 and v exhibit a much lower sensitivity. Table 4 shows the r 2 coefficients, the objective function J 1/2 and the Mean Bias Error (MBE), computed from the experimental vs. simulated C/C0 series shown in Figure 9. The MBE is calculated accounting for experimental imprecision as ( 14)

whereby the modelling error is calculated as the difference between the modelled concentration signal cB and the closest bound of the experimental concentration interval . All r 2 coefficients are larger than 0.99, even for the Q200 setting. The values for J 1/2 (that is analogous to a Root Mean Squared Error) is smaller than 1.6×10 -2 in all 21 experiments, and about 5×10 -3 for most of them. Lastly, the MBE is smaller than 1%. For all three indicators, the Q200 setting does not significantly poorer values than for the other flow rates. For instance, for MHPM A, the r 2 for Q200 is similar to that of Q10, its MBE is smaller and its J 1/2 is very similar. Similar remarks can be made for MHPM B and MHPM C. This confirms that the behaviour of the model is satisfactory, not only on the average, but also for all individual experiments for all three MHPM.

Discussion

The linearity of the exchange coefficient k12 and the dispersion coefficients Di with respect to the flow rate under laminar flow was expected from theoretical considerations [START_REF] Majdalani | Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments[END_REF], because these coefficients reflect the statistics of the fluctuations of the velocity field about the mean values. Such fluctuations are expected to be proportional to the average flow velocity (thus to the flow rate) when the flow rate is small. The present experimental and modelling results confirm the theoretical assumption of a linear relation between the flow rate and the mass exchange coefficients over a wide range of flow rate values, even for a strongly heterogeneous MHPM. This is also true for the flow velocities and dispersion coefficients in the two regions of the model. The volume fractions of the two regions and the partition coefficient of the flow rate between these regions are found to be flow rate independent. As a consequence, the flow velocity is proportional to the flow rate and the dispersivity is flow rate independent in both regions. We note that e.g. Hansen ( 2022) also inferred a flow velocity proportional to the flow rate and a flow rate independent dispersivity using a very different experimental setup and a different model, based on fractional mass exchange between the mobile and immobile regions. While [START_REF] Hansen | Experimental support for a simplified approach to CTRW transport models and exploration of parameter interpretation[END_REF] concludes that the exchange coefficient is flow rate independent, we obtain a linear relationship between k12 and the flow rate. These two findings are not necessarily contradictory in that the structures of the two exchange models are different. The three MHPMs used in the present study were built following the same geometric specifications. They are identical within a geometric precision of 1 mm (for a total length of 150 mm). This may be considered an excellent geometric replicability. Comparing the calibration results in Tables 2 and3 gives an idea of the uncertainty in the parameters with respect to MHPM manufacturing for almost indiscernible geometries. While the volume and flow distribution parameters i and i in Table 2 are remarkably stable from one MHPM to the next, the dispersion coefficients in both regions vary by 150% and the exchange coefficient varies by more than 300%. Dropping the largest discharge from the experimental set contributes to increase the consistency of the calibration results: all model parameters, including the exchange coefficient k12, are remarkably consistent for all MHPMs. This seems to indicate that the highest flow rate (Q200 pump setting) is the limit of applicability of the flow rate independent model formulation. For this flow rate, the residence time of the tracer within the MHPM is approximately 23 s, which corresponds to an average flow velocity of 6.6 mm/s. Assuming that all the flow passes through the 2 cm high central conduit yields an upper bound Re = 132 for the Reynolds number. Turbulence effects are therefore an unlikely cause. However, it is suspected that, owing to the strongly contrasted MHPM geometry, the effective flow paths are not exactly the same for the Q200 setting as for smaller flow rates, thus inducing different model calibration results. In order to compare our modelling results with previous literature findings, namely [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF] and [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF], we recall that in the present study u varies between 0.2 and 20 cm/min. [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF] found a linear relation between u and the mass exchange coefficient k12 for 0.3 < u < 2.7 cm/min, but for 0.05 < u < 0.1 cm/min the relation between k12 and u was rather quadratic (k12 = 94.23 × u 2.3 ). In both cases, the fit of the k12-u relationship was based on 3 data points, for which the experimental confidence interval is not known. Thus, according to [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF], the k12-u relationship is quadratic at low pore water velocity and linear at higher velocities. We argue that more than 3 data points are needed to deduce a good fit and this is why we use 7 flow rates in our study. In our study, the k12-u relationship is linear, in agreement with [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF] for high velocities, and even beyond the range of velocities explored by [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF]. [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF] tried to fit a relation in the form to their data and evaluated the fitting quality using the coefficient of determination r 2 . Since several, different experimental conditions (column material, column depth, column diameter, velocity range) were involved, [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF] obtained several values for the power b: b = 2.15 (r 2 = 0.965) for 0.05 < u < 0.4 cm/min, b = 1.79 (r 2 = 0.971) for 0 < u < 0.8 cm/min, b = 1.3 for 0 < u < 6 cm/min (r 2 = 0.954). In the latter case, a linear relation with b = 1 was also possible with r 2 = 0.933. [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF] concluded that k12-u relationship follows a power law with a power coefficient b that ranges from 1 to 2 where k12 scales linearly with u at low pore water velocity, and k12 may assume any value between 0.5 and 2 at higher velocities (which contradicts the conclusions of [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF]).

In the review of [START_REF] Maraqa | Prediction of mass-transfer coefficient for solute transport in porous media[END_REF], that is based on 19 experiments, the k12-u relationship is found to be sublinear (b < 1) and the value of b depends on the type of porous medium: b = 0.85 (r 2 = 0.57) for a non-aggregated medium with 0.01 < u < 0.12 cm/min, b = 0.71 (r 2 = 0.52) for an aggregated medium with 0.01 < u < 0.12 cm/min, and b = 0.76 (r 2 = 0.7) for a stony medium with 0.006 < u < 0.045 cm/min. The review of [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF] is based on 20 experiments. Its conclusions are that the k12-u relationship rather follows

, where b has no stable value. Table 3 in [START_REF] Griffioen | Interpretation of two-region model parameters[END_REF] shows that b can take any value between -0.3568 (experiment of [START_REF] Khan | A laboratory study of the dispersion scale effect in column outflow experiments[END_REF]) and 4.8543 (experiment of [START_REF] Nielsen | Miscible displacement in soils, Experimental information[END_REF]). The results of [START_REF] Bajracharya | Nonequilibrium solute transport parameters and their physical significance: Numerical and experimental results[END_REF], [START_REF] Maraqa | Prediction of mass-transfer coefficient for solute transport in porous media[END_REF][START_REF] Griffioen | Interpretation of two-region model parameters[END_REF] show that, even though the relationship k12-u is expected to be linear for theoretical reasons, experimental data do not generally reflect such linearity. However, such lack of linearity may stem from various sources. For instance, solute density effects may be present under vertical flow conditions and induce a non-linear behaviour in the k12-u relationship. The lack of linearity may also be explained by the absence of replicates. Not replicating the measurements precludes the width of the experimental confidence interval to be assessed, thus making it impossible to determine whether a linear (or another type of) law would pass through the experimental error boxes. We consider that our experimental data set is reliable because the absence of solute density effects has been verified experimentally and because tehere are 9 replicates for each flow rate: 3 replicates of each experiment and 3 replicates of the porous medium. A last discussion point is how the conclusions of the present work, carried out using a single, strongly contrasted heterogeneity, can be transposed to more natural aquifers, involving much larger scales and less contrasted heterogeneities. The present MHPMs were designed so as to maximize the contrast between the fast and slow flow paths. The purpose was to make the experimental transport process as difficult as possible to model using the standard MRAD model. From the authors' experience on MHPMs placed in series [START_REF] Majdalani | Solute transport in periodical heterogeneous porous media: Importance of observation scale and experimental sampling[END_REF], increasing the number of heterogeneities tends to smooth out the transport process, bringing it closer to a Fickian behaviour. This is nothing but a consequence of the central limit theorem (see e.g. [START_REF] Klafter | Stochastic pathway to anomalous diffusion[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. Decreasing the contrast between the permeabilities of the two media or making the volume occupied by one of them significantly smaller than the other also contributes to decrease the contrast between the slow and fast transport fluxes, which also brings the BTCs closer to those of a single transport mode, Fickian behaviour. That the reformulated MRAD model reproduces satisfactorily the BTCs of a single, highly contrasted MHPM is therefore seen as an encouraging sign for the modelling of more natural aquifers containing multiple heterogeneities. Obviously, this conclusion holds only provided that the properties (size, porosity, permeability) of the heterogeneities do not exhibit large scale variations that would be impossible to capture by a model with uniform parameters.

Conclusion

In this paper we studied solute transport in heterogeneous model porous medium under a wide range of flow rates. We conducted tracer experiments with seven tested flow rates ranging from 0.25 to 25 l/h, that is over two orders of magnitude. In comparison to other studies where the porous medium is moderately heterogeneous, our experiment is characterized by a high degree of heterogeneity. Solute transport experiment were carried out under horizontal flow and so that no solute density effects are present in the MHPM. In order to make our experimental data reliable for modelling exercise, all tracer step experiments were done with replicates: 3 replicates of each experiment and the ensemble of the experimental protocol was replicated on 3 identical MHPMs (A, B, and C). The experimental breakthrough curves show a double 'S' shape behaviour in accordance with other literature findings, and that the 'S' shape is a stable behaviour that does not vary with flow rate under a suitable change of variable on the abscissa axis: ω = Qt/V0. This shows that there exists a model with flow-rate independent parameters that can reproduce all experimental BTCs. The BTCs are modelled successfully using a reformulated version of the MRAD model with only two mobile regions. The new formulation enforces mass balance for both water and the solute, and uses flow rate independent parameters. The simulated BTCs are in close agreement with the experimental ones, with an error around 3%, r 2 coefficients above 0.99 and a Mean Bias Error smaller than 10 -2 . This validates the assumption of linear flow velocities, dispersion coefficients and exchange coefficients with respect to the flow rate in the two region model. Although the assumption of linearity between transport parameters and the flow rate has been suggested in previous studies, no clear experimental evidence for this can be found in the literature. Our study can be considered to provide a clear experimental evidence of the linearity assumption between MRAD parameters and flow rate. The variability of the model parameters from one MHPM to the other is strongly reduced when the largest flow rates are dropped from the experimental data sets. This may be an indication that the largest flow rate is at the limit of model applicability. Our experimental data set is deemed reliable owing to its high sampling rate, the use of several replicates for each experiment, all done on several replicates of the MHPM. We should like to conclude with a remark on the one-dimensional nature of the transport model. Although MHPMs (and the resulting flow within them) are three-dimensional, most experiments performed in the literature (including the present ones) involve one-dimensional flow models. These are upscaled models (Farmer, 2002) in that they attempt to provide a onedimensional representation of the transport process on the (low) resolution of the MHPM size, while the actual transport process is spatially variable and three-dimensional on a much smaller scale. Being one-dimensional, the model includes a single (longitudinal) dispersion coefficient. However, in real-world situations, dispersion involves a tensor formulation with three principal directions. A logical next step in the present research should be to investigate of how the MRAD model can be generalized to two-and three-dimensional modelling and whether the rate of convergence to the Fickian limit and the linear dependence to the flow is preserved when the MHPM is not only strongly heterogeneous but also strongly anisotropic in two and three dimensions. 7.05 × 10 -4 10 -3 1.47 × 10 -3 1.06 × 10 -3 3.9 × 10 -4 J 1/2 (-)

List of Tables

2.66 × 10 -2 3.17 × 10 -2 3.84 × 10 -2 3.22 × 10 -2 5.9 × 10 -3 av 1.56 × 10 -2 0.76 × 10 -2 0.56 × 10 -2 0.96 × 10 -2 5.3 × 10 -3 1.63×10 -3 0.994 1.07×10 -2 1.13×10 -3 0.991 2.35×10 -3 7.74×10 -4 Q4 0.998 4.33×10 -3 1.55×10 -3 0.999 8.44×10 -3 -2.17×10 -4 0.997 4.34×10 -3 -1.71×10 -3 Q10 0.995 4.89×10 -3 3.51×10 -3 0.997 7.01×10 -3 8.39×10 -3 0.999 3.66×10 -3 9.61×10 -4 Q20 0.999 4.03×10 -3 3.74×10 -4 0.998 1.23×10 -2 9.05×10 -3 0.997 2.53×10 -3 1.23×10 -3 Q40 0.999 2.87×10 -3 3.04×10 -4 0.999 6.33×10 -3 4.83×10 -3 0.999 4.80×10 -3 2.11×10 -3 Q100 0.999 1.25×10 -2 -1.61×10 -3 0.995 8.15×10 -3 7.13×10 -4 0.997 1.83×10 -3 6.18×10 -5 Q200 0.995 5.04×10 -3 2.52×10 -3 0.999 1.59×10 -2 -1.05×10 -3 0.992 1.08×10 -2 5.17×10 -3 The variable change ω = Qt/V0 is adopted for the abscissa axis. The three digit figures in the legend refer to the peristaltic pump setting, see Table 1 for the flow rate values. Figure 9. Experimental vs. modelled breakthrough curves for the three MHPM (A, B, and C) and for the seven tested flow rates.

  (2018)), whereby the flow velocities and volume fractions are calibrated without constraining the conservation of water flux (let alone solute fluxes). The flow is divided into R regions flowing in parallel. Let Ai and Qi be respectively the crosssectional area and discharge in Region i. They are related to the total area A and discharge Q with (6) Assuming discharge-proportional dispersion and exchange coefficients, the governing equation in Region i is (7)

Figures Figure 1 .

 1 Figures

Figure 3 .

 3 Figure3. Model heterogeneous porous medium Y2 where the heterogeneity (white rectangle) is located in the center of one face. As in Figure1, dashed area represents 1 mm glass beads and white area represents the heterogeneity (rectangular pore).

Figure 4 .

 4 Figure 4. Tracer step experiment setup.

Figure 7 .

 7 Figure 7. Step experiments with flow rate Q100. Top: Column Y1 in Up and Down positions. Bottom: Column Y2 in Up and Down positions.

Figure 8 .

 8 Figure8. Experimental breakthrough curves for columns MHPM A, B, and C. The variable change ω = Qt/V0 is adopted for the abscissa axis. The three digit figures in the legend refer to the peristaltic pump setting, see Table1for the flow rate values.

Table 1 .

 1 Values and symbols of the flow rates tested in step experiments on columns Xi (i = 1 to 3) also called MHPM A,

	MHPM B, and MHPM C
	Symbol	Flow rate (l/h)	Experiment duration (min)	Sampling Interval (s)
	Q2	0.25	180	10
	Q4	0.5	90	10
	Q10	1.25	40	5
	Q20	2.5	20	5
	Q40	5	10	1
	Q100	12.5	3	1
	Q200	25	2	1

Table 2 .

 2 Calibration results with all flow rates. The average and standard deviations in each row are computed row from the three values obtained from MHPMs A-C.

	Parameter	MHPM A	MHPM B	MHPM C	Average	Standard dev.
	α1 (cm)	2.36	2.33	1.72	2.14	0.36
	α2 (m)	0.373	0.269	0.262	0.301	0.062
	μ1 (-)	0.294	0.288	0.270	0.284	0.012
	ν1 (-)	0.598	0.546	0.593	0.579	0.029
	k12 (m -1 )	0.578	0.295	0.948	0.607	0.327
	J (-)

Table 3 .

 3 Calibration results without Q200. The average and standard deviations in each row are computed row from the three values obtained from MHPMs A-C.

	Parameters	MHPM A	MHPM B	MHPM C	Average	Standard dev.
	α1 (cm)	1.55	1.15	1.48	1.39	0.21
	α2 (cm)	0.334	0.291	0.283	0.303	0.024
	μ1 (-)	0.271	0.228	0.264	0.254	0.023
	ν1 (-)	0.592	0.506	0.591	0.563	0.049
	k12 (m -1 )	0.898	0.707	0.823	0.809	0.096
	J (-)	5.78 × 10 -4	6.86 × 10 -4	8.19 × 10 -4	6.94 × 10 -4	1.21 × 10 -4
	J 1/2 (-)	2.40 × 10 -2	2.62 × 10 -2	2.87 × 10 -2	2.63 × 10 -2	2.4 × 10 -3
	av	1.68%	0.72%	0.51%	0.97 × 10 -2	6.3 × 10 -3

Table 4 .

 4 Goodness-of-fit indicators for experimental vs. modelled C/C0.

	Flow rate		MHPM A			MHPM B			MHPM C
		r 2	J 1/2	MBE	r 2	J 1/2	MBE	r 2	J 1/2	MBE
	Q2	0.998	5.16×10 -3

Table 5 .

 5 Dimensionless numbers for similarity analysis. The Reynolds number is computed as and the Peclet number in region k as .

	MHPM	Re (-)	Pe1 (-)	Pe2 (-)
	A	1.32 -132	9.7	44.9
	B	1.32 -132	13.0	51.5
	C	1.32 -132	10.1	53.0

Acknowledgements

The authors thank Jean-Philippe Chazarin from the laboratory HydroSciences Montpellier -France for his help in developing the electronic devices used in the experiments.

Appendix. Numerical solution of the transport equations

The present appendix details the numerical solution of the transport equation (1a), that can be rewritten in the following form under the assumption of space-independent coefficients in the equation

The equation is solved using a three step time splitting procedure (Strang, 1968). To do so, Eq. (A.1) is split into three equations:

(A.2a)

where Eqs. (A.2a-c) are solved numerically using the same computational time step t, and each equation (A.2b-c) uses the result of the previous one as a starting point over the time step. Equation (A.2a) is solved using a finite volume procedure. Within each region i (the subscript i is dropped in the notation for the sake of readability), the following formula is used:

where is the average value for c in the kth cell at time step n, and is the average value of c at the interface between Cells k-1 and k between time levels n and n+1. The salient point of the algorithm lies in the definition of the interface values . In order to minimize numerical diffusion, we used an arbitrary time step explicit method presented in [START_REF] Leonard | Note on the von Neumann stability of explicit one-dimensional advection schemes[END_REF]. This method removes the restriction on the time computational time step by allowing the dimensionless CFL number to take any arbitrary large value. t is set up to the largest possible value allowing the fastest region to travel over the length L of the computational domain within t, that is


M.Sc Camille Perchoux
email: camille.perchoux@gmail.com

Ph.D Basile Chaix

M.Sc Ruben Brondeel

Ph.D Yan Kestens

Residential buffer, perceived neighborhood, and individual activity space: new refinements in the definition of exposure areas -The RECORD Cohort Study

Keywords: Environmental exposure, Residential buffer, Perceived residential neighborhood, Activity space, Selective daily mobility bias

come

Residential buffer, perceived neighborhood, and individual activity space: New refinements in the definition of exposure areas -The RECORD Cohort Study

INTRODUCTION

The previous decades have witnessed a renewed focus on the effect of environmental factors on population health. More recently, technological developments in the collection of locational data and advances in spatial analytic methods have opened up new possibilities to observe and analyze space-time exposures and go beyond the commonly used residential neighborhood [START_REF] Chaix | Geographic Life Environments and Coronary Heart Disease: A Literature Review, Theoretical Contributions, Methodological Updates, and a Research Agenda[END_REF][START_REF] Diez Roux | Investigating neighborhood and area effects on health[END_REF][START_REF] Kerr | Exposure Exposed[END_REF][START_REF] Pickett | Multilevel analyses of neighbourhood socio-economic context and health outcomes: a critical review[END_REF][START_REF] Riva | Toward the next generation of research into small area effects on health: a synthesis of multilevel investigations published since July 1998[END_REF]. Yet, such advances call into question which environments or exposure areas are relevant.

Commonly used definitions of the exposure area in place and health studies include administrative neighborhoods (i.e. census tracts, postal codes) and residence-centered circular or street network buffers areas [START_REF] Leal | The influence of geographic life environments on cardiometabolic risk factors: a systematic review, a methodological assessment and a research agenda[END_REF]. More recently, participants' perceived residential neighborhoods have been proposed as an alternative [START_REF] Chaix | Neighbourhoods in eco-epidemiologic research: delimiting personal exposure areas. A response to Riva, Gauvin, Apparicio and Brodeur[END_REF]Vallée et al., 2014;[START_REF] Vallée | EXAMINATION OF HOW NEIGHBORHOOD DEFINITION INFLUENCES MEASUREMENTS OF YOUTHS'ACCESS TO TOBACCO RETAILERS: A METHODOLOGICAL NOTE ON SPATIAL MISCLASSIFICATION[END_REF]. The perceived residential neighborhood relies on the participants' cognitive construct of their neighborhood [START_REF] Coulton | Mapping Residents' Perceptions of Neighborhood Boundaries: A Methodological Note[END_REF][START_REF] Guest | How urbanites define their neighborhoods[END_REF]. Substantial differences in the measures of environmental exposures (park availability, commercial physical activity facilities, restaurants, and food stores) were observed between the residential and the perceived neighborhood [START_REF] Colabianchi | Adolescent self-defined neighborhoods and activity spaces: Spatial overlap and relations to physical activity and obesity[END_REF].

These definitions of the exposure area are however exclusively focused on the residential neighborhood, even if individuals are mobile and get exposed within a variety of environments [START_REF] Chaix | Neighbourhoods in eco-epidemiologic research: delimiting personal exposure areas. A response to Riva, Gauvin, Apparicio and Brodeur[END_REF][START_REF] Matthews | Spatial Polygamy and the Heterogeneity of Place: Studying People and Place via Egocentric Methods[END_REF][START_REF] Matthews | Spatial Polygamy and Contextual Exposures (SPACEs): Promoting Activity Space Approaches in Research on Place And Health[END_REF][START_REF] Perchoux | Conceptualization and measurement of environmental exposure in epidemiology: Accounting for activity space related to daily mobility[END_REF]Shareck et al., 2014a;Shareck et al., 2014b). Exposure outside the residential neighborhood might differ from exposure within the residential neighborhood [START_REF] Basta | Neighborhoods, daily activities, and measuring health risks experienced in urban environments[END_REF][START_REF] Inagami | Non-residential neighborhood exposures suppress neighborhood effects on self-rated health[END_REF][START_REF] Kestens | Association between Activity Space Exposure to Food Establishments and Individual Risk of Overweight[END_REF][START_REF] Kestens | Using experienced activity spaces to measure foodscape exposure[END_REF][START_REF] Lipperman-Kreda | Youth activity spaces and daily exposure to tobacco outlets[END_REF][START_REF] Mason | Attributing activity space as risky and safe: The social dimension to the meaning of place for urban adolescents[END_REF][START_REF] Setton | The impact of daily mobility on exposure to traffic-related air pollution and health effect estimates[END_REF][START_REF] Zenk | Activity space environment and dietary and physical activity behaviors: A pilot study[END_REF]. Most studies accounting for exposure beyond the residential neighborhood have used the concept of activity space -i.e. the set of daily visited activity locations [START_REF] Golledge | Spatial Behavior[END_REF] -to operationalize personal areas of exposure.

Despite the growing use of the concept of activity space, few studies were able to report how much residential and non-residential environmental exposures differ. For instance, a study based on the tracking of participants with GPS receivers in the Seattle area observed that more than 90% of the built environment measures differed between residential and non-residential locations [START_REF] Hurvitz | Home Versus Nonhome Neighborhood: Quantifying Differences in Exposure to the Built Environment[END_REF]. Similar differences were found elsewhere [START_REF] Basta | Neighborhoods, daily activities, and measuring health risks experienced in urban environments[END_REF][START_REF] Crawford | Conceptualizing and comparing neighborhood and activity space measures for food environment research[END_REF][START_REF] Kestens | Using experienced activity spaces to measure foodscape exposure[END_REF][START_REF] Lipperman-Kreda | Youth activity spaces and daily exposure to tobacco outlets[END_REF]Shareck et al., 2014b;[START_REF] Zenk | Activity space environment and dietary and physical activity behaviors: A pilot study[END_REF]. Except one study [START_REF] Crawford | Conceptualizing and comparing neighborhood and activity space measures for food environment research[END_REF], these published reports have paid no attention to the difference in built environment measures between the perceived residential neighborhood and the broader activity space.

Whereas reducing measurement error in exposure measures has been recommended, accounting for the non-residential places visited does also require caution. Concerns about confounding related to the selective daily mobility bias have been raised [START_REF] Chaix | An interactive mapping tool to assess individual mobility patterns in neighborhood studies[END_REF][START_REF] Chaix | GPS tracking in neighborhood and health studies: A step forward for environmental exposure assessment, a step backward for causal inference[END_REF][START_REF] Kerr | Exposure Exposed[END_REF][START_REF] Kestens | Association between Activity Space Exposure to Food Establishments and Individual Risk of Overweight[END_REF]. This bias arises when "measures of accessibility to given environmental resources are also determined from the locations that were specifically visited to use the corresponding resources" (Chaix et al., 2013, p.48). To overcome this potential source of confounding, Chaix et al. suggested to either exclude the activity places visited related to the behavior of interest when measuring exposure or to only retain major activity locations (Hägerstrand, 1970) corresponding to constrained activities that cannot be rescheduled or carried out in another location [START_REF] Chaix | An interactive mapping tool to assess individual mobility patterns in neighborhood studies[END_REF]. Such a selection of activity places could provide measures of exposure that mitigate the selective daily mobility bias.

Objectives

This study aimed to investigate how environmental exposures vary according to the spatial definition of the exposure area, using two built environment characteristics conducive to walking, i.e. the density of destinations and green spaces [START_REF] Chaix | The environmental correlates of overall and neighborhood based recreational walking (a cross-sectional analysis of the RECORD Study)[END_REF][START_REF] Sugiyama | Destination and Route Attributes Associated with Adults' Walking: A Review[END_REF]. A first objective was to evaluate how exposure measurements varied depending on whether exposure areas were defined as i) street-network residential buffers, ii) self-reported perceived residential neighborhoods, or iii) activity space areas. Variations in exposure measures according to the urbanicity of the residence were compared according to these three spatial definitions. A second objective was to evaluate the selective daily mobility bias. To do so, we assessed whether activity space exposure measures and gradients differed when using either i) all reported destinations (full activity space) or ii) only destinations unrelated to the exposure of interest (truncated activity space, theoretically reducing the selective mobility bias).

METHODS

Population

The study relies on the second wave of the RECORD Cohort Study [START_REF] Chaix | Cohort Profile: Residential and non-residential environments, individual activity spaces and cardiovascular risk factors and diseases-The RECORD Cohort Study[END_REF]. Overall, 5,542 participants were surveyed without a priori sampling (convenience sample) between

Activity space data

Self-reported activity places were geocoded using the VERITAS application ('Visualization and Evaluation of Regular Individual Travel destinations and Activity Spaces') [START_REF] Chaix | An interactive mapping tool to assess individual mobility patterns in neighborhood studies[END_REF]. The VERITAS application is a web-based computer tool that integrates Google Maps interactive mapping functionalities, and allows users to self-report activity locations and perceived spaces (Chaix et al., 2012, p. 441). With help of a survey technician, participants geocoded their regular activity locations, and provided the frequency of visit to these locations (coded as at least once per month, once per week, or more often). Participants were successively asked to locate their places of residence, workplaces, services (supermarket, outdoor market, bakeries, butchers, fruit and vegetables shops, specific food stores, tobacco/press shops, bank, post offices, etc.), transportation stations used from home, recreational activities (sports facilities, place of cultural activity, place of labor organization, political, or religious activity), and social activities (place of social activities, place where participants take relatives, place where participants visit relatives). The participants were further asked to draw the boundaries of their perceived neighborhood. More details on the VERITAS data collection in the RECORD Study can be found in Chaix et al. (2012b).

Spatial definitions of exposure areas

'Classical' residential exposure area: A 1000 meter street network buffer was defined around each participant's home. This distance, previously used in place and health research studies, corresponds to a 15 minute walk [START_REF] Brondeel | Use of healthcare services in the residence and workplace neighbourhood: The effect of spatial accessibility to healthcare services[END_REF][START_REF] Chaix | The environmental correlates of overall and neighborhood based recreational walking (a cross-sectional analysis of the RECORD Study)[END_REF][START_REF] Frank | Linking objectively measured physical activity with objectively measured urban form: findings from SMARTRAQ[END_REF][START_REF] Karusisi | Spatial accessibility to specific sport facilities and corresponding sport practice: the RECORD Study[END_REF][START_REF] Troped | The Built Environment and Location-Based Physical Activity[END_REF][START_REF] Villanueva | The impact of neighborhood walkability on walking: Does it differ across adult life stage and does neighborhood buffer size matter?[END_REF].

Perceived residential exposure area: The perceived residential neighborhood was obtained from participants' self-report drawing the perceived boundaries of their neighborhood using VERITAS.

Activity space exposure areas: Activity space exposure areas were defined using buffers around activity destinations. A full activity space included all regular activity places reported in VERITAS. In order to control for the selective daily mobility bias, a truncated activity space was further defined, by removing the activity places specifically referring to the activities related to the exposure of interest. For example, the spatial accessibility to services was measured from all activity locations reported except from services themselves. It means that only relatively constrained and fixed destinations were retained, including the residence, the workplace, the regular bank, and the places where participants take relatives. The spatial accessibility to green spaces was measured from all activity locations except from sport activity destinations, as such destinations include the green and open spaces that were specifically visited to exercise.

For both definitions of the activity space, street network buffers were constructed around each reported activity location. Because the degree of exposure plausibly depends on the time spent at the location or on the frequency of visit, varying buffer radiuses were used depending on the types of activity locations. As we had no data on the time spent around each location, it was decided to use larger buffers for major activity locations such as the residence and the workplace (1000 m), intermediary buffers for recreational/social activity locations (500 m), and smaller buffers for service activity locations (200 m) [START_REF] Chaix | An interactive mapping tool to assess individual mobility patterns in neighborhood studies[END_REF]. This hierarchy of buffer sizes has been used previously in a study examining built and social environment influences on recreational walking [START_REF] Perchoux | Accounting for the daily locations visited in the study of the built environment correlates of recreational walking (the RECORD Cohort Study)[END_REF]. Figure I illustrates the residential buffer, perceived neighborhood, and full activity space as areas of exposure.

Environmental data

Two exposure measures that have been related to walkability were computed for each of the four area definitions: the proportion of green space and the density of destinations (number per km²). The location of green spaces was obtained from a 2008 geographic layer of the Institute of Urban Planning of the Ile-de-France Region (IAU-IDF) on public parks and green spaces. The location of the residence in the Paris Ile-de-France region was examined as a proxy of urbanicity (City center, inner suburbs, and outer suburbs). This categorization of urbanicity is based on a recognized and official administrative subdivision of the Ile-de-France region, previously used to look at spatial dynamics in the Ile-de-France region [START_REF] Charreire | Identifying built environmental patterns using cluster analysis and GIS: Relationships with walking, cycling and body mass index in French adults[END_REF][START_REF] Feuillet | Spatial heterogeneity of the relationships between environmental characteristics and active commuting: towards a locally varying social ecological model[END_REF][START_REF] Perchoux | Assessing patterns of spatial behavior in health studies: Their socio-demographic determinants and associations with transportation modes (the RECORD Cohort Study)[END_REF]. It distinguishes the city of Paris itself, the first crown of counties ("Départements") around Paris, and the second crown of counties.

Statistical analyses

Analyses of variance were used to examine variations in the size of the street-network residential buffer, the perceived residential neighborhood, and the full and truncated activity spaces in relation to age, sex, income, urbanicity of the residence, proportion of green space, and density of destinations. Paired sample t-tests were used to assess differences in exposure measures between the different definitions of exposure areas. Jonckheere-Terpstra (JT) tests were performed to assess trends in exposure between ordered classes. All analyses were conducted with SAS version 9.2.

RESULTS

Description of the study sample

From the initial sample of 5542 participants, we excluded 55 participants living and 996 participants regularly traveling outside the Ile-de-France region, and 108 participants regularly visiting a secondary home. Two participants were further excluded due to missing sociodemographic data. The final sample included 4,383 individuals with a mean age of 53 years (range: 32-85), predominantly male (67%), French (87%), and with a stable employment status (56%). Of our sample, 26% lived in Paris City, 46% in the inner, and 27% in the outer suburbs.

Participants reported a median number of 13 distinct activity locations (range: 2-42). The median size of the exposure area was of 1.8 km 2 (0.5-2.9) for the street-network residential buffers, 0.5 km 2 (range: 0.0-277.5) for the perceived residential neighborhood, 3.8 km 2 (1.1-11.4) for the full activity space, 3.7 km 2 (range: 1.0-10.8) for the sport places-truncated activity space, and 3.1 km 2 (range: 0.5-10.0) for the services-truncated activity space. Unadjusted relationships between individual/environmental characteristics and the size of the different exposure areas are reported in Appendix I. Sizes of both the street-network residential buffer and activity space increased with urbanicity (possibly due to higher street connectivity in more urban environments). A particularly strong positive association was documented between the proportion of green spaces in the area and the size of the perceived residential neighborhood (suggesting that participants located close to green spaces may extend their perceived neighborhood so as to include a greater surface of green spaces in it).

Differences in environmental exposure by neighborhood definition

Table I shows average values of environmental measures by exposure areas, and Table II differences between the different types of exposure areas.

Overall, the proportion of green space in the exposure area was larger in the outer suburbs than in the city center (Table I). Regardless of the exposure area, the density of services in the exposure area increased with urbanicity.

Street-network residential buffer vs. perceived residential neighborhood

The mean proportion of green space was higher in the perceived residential neighborhood (0.080) than in the street-network residential buffer (0.071) (Table I). This difference increased from the inner suburbs to the city center (p<0.001) (Table II).

Again, the density of destinations was greater in the perceived residential neighborhood than in the street-network residential buffer (Table I). The observed difference of exposure between these areas revealed an increasing trend from the outer suburbs (59.4) to the city center (127.5) (p<0.001) (Table II).

Street-network residential buffer vs. truncated activity space

Regarding the proportion of green space, no overall differences were found between the streetnetwork residential buffer and the truncated activity space (Table II). However, participants living in the center had higher exposure to green spaces in their truncated activity space than in their street-network residential buffer, while the contrary was true for suburbanites (p<0.001).

Overall, the truncated activity space contained a higher density of destinations (441.1) than the street-network residential buffer (360.1) (Table I). However, while individuals living in the city center had a higher density of destinations in their street-network residential buffer (907.8) than in their truncated activity space (868.1), outer suburbanites had a 2.4 times higher density of services in their truncated activity space than in their street-network residential buffer (Table I

& II).

Perceived residential neighborhood vs. truncated activity space

The proportion of green spaces and the density of destinations were lower in the truncated activity space than in the perceived residential neighborhood (Table II).

Differences in exposure to green spaces between the two varied by urbanicity of the residence (p<0.001), with a lower exposure to green spaces in the truncated activity space than in the perceived residential neighborhood for residents of the city center or inner suburbs (-0.013) as opposed to those of outer suburbs (0.002) (Table II).

The differences in the accessibility to destinations varied by urbanicity (Table II). A strong trend (p<0.001) in the difference between the perceived residential neighborhood and the truncated activity space showed that urban individuals had a higher exposure to destinations in their perceived residential neighborhood, whereas suburbanites had a higher exposure to destinations in their truncated activity space.

Full Activity Space vs. Truncated Activity Space

The exposure estimates were higher in the full than in the truncated activity space (Table II).

Accessibility to green spaces increased from the city center to the outer suburbs, but considering the truncated rather than the full activity space slightly attenuated this gradient.

Stratification by urbanicity of the residence shows higher accessibility to destinations when considering the full rather than the truncated activity space, with more pronounced differences for suburban rather than urban residents (p<0.001) (Table II).

DISCUSSION

This study had two major objectives. First, we compared measures of environmental exposure conducive to walking between a classical street network-buffer centered on the residence, the perceived residential neighborhood, and the activity space, by urbanicity of the residence. Second, we examined to which extent selective daily mobility might result in biased estimates of environmental exposures.

Regarding the first objective, the estimates of environmental exposures differed according to the definition of the exposure area. Compared to the street network buffer, we observed higher levels of exposures in the perceived residential neighborhood, as reported before [START_REF] Colabianchi | Adolescent self-defined neighborhoods and activity spaces: Spatial overlap and relations to physical activity and obesity[END_REF]. This is likely attributable to the fact that the perceived residential neighborhood is likely shaped by the main local destinations of people, potentially the services or green spaces.

Measures of environmental exposure that account for daily mobility (full and truncated activity spaces) differed from those based on the street-network residential buffer and on the perceived residential neighborhood. Exposure to destinations was higher in the activity space than in the street-network residential buffer, in accordance with other studies on the food environment [START_REF] Basta | Neighborhoods, daily activities, and measuring health risks experienced in urban environments[END_REF][START_REF] Crawford | Conceptualizing and comparing neighborhood and activity space measures for food environment research[END_REF][START_REF] Hurvitz | Home Versus Nonhome Neighborhood: Quantifying Differences in Exposure to the Built Environment[END_REF][START_REF] Kestens | Using experienced activity spaces to measure foodscape exposure[END_REF].

Again, this is probably attributable to the fact that the participants' activity space is shaped to some extent by the visits that they make to destinations of various kinds. Finally, in most cases, differences in exposure between definitions of exposure areas showed a gradient associated with urbanicity. Such variations suggest that the measurement bias induced by the exclusive use of a given exposure area compared to another one is differential among participants living in the city center, and inner and outer suburbanites. We finally observed disparities in access to urban resources. Participants living in the city center had higher densities of destinations in their residential neighbourhood than in their activity space (i.e. accounting for non-residential destinations lowered the observed densities), whereas the contrary was true for suburban participants.

Regarding the second objective, we observed no clear difference in the gradient of exposure according to urbanicity between the truncated activity space and the full activity space. However, additional analyses by household income (Appendix II) showed that exposure to green spaces increased with household income when considering the full activity space, while the trend was absent with the truncated activity space. A potential explanation is that despite the fact that people of different socioeconomic background have comparable accessibility to green spaces from their core daily activity locations, high-income participants are more likely to regularly visit parks than low-income participants. This hypothesis is consistent with current literature that reported a positive relationship between household income and park visitation frequency [START_REF] Mowen | Change and stability in park visitation constraints revisited[END_REF][START_REF] Scott | Perceived Constraints to Park Usage[END_REF]. Therefore, as previously emphasized [START_REF] Chaix | An interactive mapping tool to assess individual mobility patterns in neighborhood studies[END_REF][START_REF] Chaix | GPS tracking in neighborhood and health studies: A step forward for environmental exposure assessment, a step backward for causal inference[END_REF], our expectation is that the exposure to green spaces assessed in the full activity space may generate bias when associated with health outcomes (e.g., walking). Using the truncated activity space may be useful to mitigate the selective daily mobility bias.

Toward refinements in the definition of exposure areas

Our findings illustrate that the spatial definition of the exposure area has a strong impact on exposure estimates and thus potentially on estimations of contextual effects on health. Although exploratory, this paper advances three conceptual and methodological points. First, accounting solely for the residential neighborhood may provide an incomplete and somewhat biased understanding of environmental effects on health behaviors. Future research should thus move toward a more comprehensive spatial definition of the exposure area that considers individuals' mobility patterns.

Second, the perceived residential neighborhood is markedly different from the activity space and from the street-network residential buffer, both in terms of size and related exposure. The higher levels of spatial accessibility to resources in the perceived residential neighborhood combined with its smaller size denotes its anisotropic character. In short, it avoids low density areas where individuals do not go, compared to the isotropic and non-discriminant streetnetwork buffer [START_REF] Chaix | Neighbourhoods in eco-epidemiologic research: delimiting personal exposure areas. A response to Riva, Gauvin, Apparicio and Brodeur[END_REF]. Further research investigating the sociodemographic, psychological, and environmental determinants of the shape of the perceived residential neighborhood is needed. However, since the perceived residential neighborhood as a cognitive construct may not exactly match the local area to which individuals are exposed [START_REF] Chaix | Neighbourhoods in eco-epidemiologic research: delimiting personal exposure areas. A response to Riva, Gauvin, Apparicio and Brodeur[END_REF], future research should elaborate the conceptual grounds for relying on a subjective rather than objective definition of spatial exposure areas.

Third, while accounting for daily mobility in place and health research is increasingly recognized as a step forward in the specification of environmental exposures and analysis of their influence on health [START_REF] Chaix | GPS tracking in neighborhood and health studies: A step forward for environmental exposure assessment, a step backward for causal inference[END_REF][START_REF] Kestens | Comments on Melis et al. The Effects of the Urban Built Environment on Mental Health: A Cohort Study in a Large Northern Italian City[END_REF], further attention should be paid to potential selection biases in studies of mobility-based measures of exposures and health. Sophisticated methods using GPS data have been proposed in order to address the selective daily mobility bias, as accounting for trip chaining and how a specific behavior in a specific place and time might be influenced by the "structure of opportunities observed at the previous or next activity location" [START_REF] Chaix | GPS tracking in neighborhood and health studies: A step forward for environmental exposure assessment, a step backward for causal inference[END_REF]Thierry et al., 2013, p. 3).

Strengths and limitations

The definition of the activity space used in this study encompassed areas around all reported destinations, including the residential neighborhood. It made it difficult to assess whether participants actually compensated for the lack of resources in their residential neighborhood by visiting activity places in their non-residential environments. The definition of the truncated activity space might also be questioned in relation to the specific locations excluded to mitigate the selective daily mobility bias. The methodological choice to remove sport activity destinations to define a truncated activity space for assessing green space exposure might not have allowed us to correctly exclude the green spaces visited by purpose by the participants as we intended to do. On the one hand, sport activity locations can take place both inside and outside, and on the other hand, social activities such as picnics could be organized in parks, and should also be removed from the exposure estimates. The present study illustrates that truncating the activity space is a particularly straightforward strategy when applied to specific environmental exposures such as green spaces or fast food restaurants and when such places can be reliably identified among the participant's visited places.

The lack of temporal information did not allow us weighting the environmental exposures by the time spent at each activity location. The mismatch in the source year of green spaces exposure data compared to individual data might have introduced a potential misestimation of the actual exposure. Finally, we might have introduced a selection bias by excluding the participants traveling outside the Ile-de-France Region, since these participants have significantly higher educational attainment and income. Thus, socio-economic disparities in exposure differences are potentially under or over-estimated, if the exposure itself had a causal effect on participation.

The strengths of the study include a large sample geographically dispersed in the whole metropolitan area of Paris, with precise geographical information of participants' activity places and perceived neighborhood boundaries that were collected through the use of an interactive mapping application. Our study is also one of the first to properly address concerns related to the selective daily mobility bias.

CONCLUSION

While more and more studies are currently collecting real-time exposure data through GPS receivers eventually in combination with environmental sensors, our study strengthens the theoretical ground for assessing multiple-place exposure by underlining the difference between residential and activity space exposures. It also sheds light on the extent to which errors in residential exposure measures may vary in magnitude according to urbanicity if one definition of the exposure area is employed rather than the one that should be ideally used. Our findings also highlight the need to address the selective daily mobility bias in health and place studies accounting for individual-level mobility. Failing to do so might lead to confounding and preclude causal inference. Future research will have to examine whether accounting for the full range of environmental exposures in a multi-place perspective provides stronger evidence on the places and populations that public health interventions should target.

February

  2011 and October 2013 during preventive health checkups conducted by the Centre d'Investigations Préventives et Cliniques (IPC) in Paris. Participants were living in one of 10 (out of 20) administrative divisions of Paris or 111 a priori selected municipalities of the Ilede-France region in 2011-2013 or had been living in these municipalities in 2007-2008 during the recruitment of the cohort (some of them had moved to other places since then). In addition to the RECORD Study inclusion criteria (residence and age 30-79 in 2007-2008), the present analyses retained only participants residing in the Ile-de-France region who reported at least one non-residential destination. The entire data collection protocol was approved by the French Data Protection Authority. All participants had to sign a consent form to enter the study.

  Destinations were obtained from the 2011 Permanent Database of Facilities of the National Institute of Statistics and Economic Studies (INSEE) and included administrations, public/private shops, health services, and entertainment facilities.

  Means and standard deviations of environmental exposures in the street-network residential buffer, the perceived residential neighborhood, and in the activity spaces by household income (n=4323)

	City center		1.921 (0.159)	1.468 (4.022)		3.980 (1.438)	3.803 (1.375)	3.222 (1.272)
	Inner suburbs		1.699 (0.224)	1.130 (2.859)		3.890 (1.408)	3.689 (1.341)	3.027 (1.225)
	Outer suburbs		1.561 (0.285)	1.615 (10.937)		3.706 (1.400)	3.487 (1.324)	2.735 (1.190)
	p value Variables	<0.001 Street-network residential	0.0878 Perceived residential	<0.001	<0.001 Full activity space	Truncated activity space <0.001 Truncated activity space
	ENVIRONMENTAL EXPOSURE buffer		neighborhood				-Destinations
	INDIVIDUAL CHARACTERISTICS Mean (SD) Proportion of green spaces, mean (SD)	JT Test p value	Mean (SD)	JT Test p	Mean (SD)	JT Test p value	Mean (SD)	JT Test p
	Age, mean (SD) High		1.667 (0.264)	2.274 (8.852)	4.011 (1.418) value	3.729 (1.334)	-	value
	30 -44 Mediun PROPORTION OF GREEN SPACES 1.724 (0.257) 1.782 (0.235)	1.094 (2.054) 1.280 (6.417)		4.188 (1.369) 4.018 (1.437)	3.988 (1.299) 3.858 (1.387)	3.347 (1.179) -
	45 -59 60 -85 Low All p value By household income a 0.071 (0.076) 1.717 (0.263) 1.721 (0.274) 1.710 (0.279) <0.001 High 0.070 (0.077)	1.519 (9.037) 1.333 (3.248) 0.502 (1.103) 0.080 (0.113) <0.001 0.078 (0.114)		4.100 (1.407) 3.180 (1.237) 3.561 (1.346) 0.080 (0.061) 3.901 (1.344) 2.978 (1.146) 3.406 (1.293) <0.001 <0.001 0.081 (0.057)	3.267 (1.224) 2.235 (0.969) 0.071 (0.053) --0.069 (0.049)
	p value Density of destinations, mean (SD) 0.713 Medium 0.071 (0.075)	0.647	0.194 0.082 (0.115)	<0.001 0.334	<0.001 0.082 (0.063)	0.001*	<0.001 0.072 (0.054)	0.443
	Sex, mean (SD) High Low	1.897 (0.168) 0.071 (0.075)	1.257 (4.401) 0.077 (0.110)		4.225 (1.476) 0.076 (0.060) -	3.487 (1.302) 0.070 (0.055)
	Female Mediun DENSITY OF DESTINATIONS 1.714 (0.271) 1.703 (0.221) All 360.1 (399.4)	1.096 (2.230) 1.071 (1.697) 461.1 (503.3)		3.724 (1.438) 3.879 (1.404) 481.2 (382.2) 3.568 (1.386) -	2.899 (1.239) 3.035 (1.549) 441.1 (407.1)
	Male Low By household income a	1.723 (0.261) 1.560 (0.275)	1.479 (7.650) 1.736 (10.053)		3.935 (1.401) 3.492 (1.269)	3.712 (1.330) -	3.048 (1.239) 2.473 (1.041)
	p value p value High	0.116 <0.001 447.3 (440.4)		<0.001 0.0147 573.7 (538.5)		0.257 <0.001	0.065 -567.4 (405.4)	1.00 <0.001 526.1 (436.5)
	Medium Income a , mean (SD) 352.7 (387.6) Low 286.8 (356.3)	<.001*	451.4 (498.4) 368.2 (455.9)	<.001*	480.6 (374.9) 402.4 (352.3)	<.001*	440.9 (396.9) 363.4 (375.3)	<.001*
	High *p < 0.05; SD: standard deviation 1.761 (0.253)	1.348 (7.144)		4.035 (1.418)	3.784 (1.348)	3.101 (1.273)
	Medium a n = 4323 when stratifying by household income 1.726 (0.265) 1.390 (7.615)		3.929 (1.447)	3.703 (1.372)	3.048 (1.258)
	Low		1.677 (0.268)	1.299 (3.701)		3.652 (1.367)	3.528 (1.325)	2.871 (1.186)
	p value		<0.001		0.929		<0.001		<0.001	<0.001
	Location of the residence, mean (SD)

a n = 4323 when stratifying by household income; SD: standard deviation Appendix II.

ACKNOWLEDGMENTS

The VERITAS project, as related to the RECORD Study, is supported by the Institute for Public Health Research (IReSP, Institut de Recherche en Santé Publique); the National Institute for Prevention and Health Education (INPES, Institut National de Prévention et d'Education pour la Santé) (Prevention Program 2007; 2010-2011 fınancial support; and 2011-2013 fınancial support); the National Institute of Public Health Surveillance (InVS, Institut de Veille Sanitaire) (Territory and Health Program); the FrenchMinistries of Research and Health (Epidemiologic Cohorts Grant 2008); the National Health Insurance Offıce for Salaried Workers (CNAM-TS, Caisse Nationale d'Assurance Maladie des Travailleurs Salariés); the Ile-de-France Regional Health Agency (ARS, Agence Régionale de Santé d'Îlede-France); the City of Paris (Ville de Paris); the Ile-de-France Regional Council (Conseil Régional d'Île-de-France, DIM SEnT and CODDIM); and the Ile-de-France Youth, Sports, and Social Cohesion Regional Direction (DRJSCS, Direction Régionale de la Jeunesse et des Sports et de la Cohésion Sociale). The VERITAS project is also supported by the Canadian Institutes of Health Research (CIHR No. TOO-105427). This work was supported by the Canadian Institutes of Health Research (Grant # 213338). RB is supported by a grant from the EHESP School of Public Health (Ecole des Hautes Etudes en Santé Publique). YK is supported by an investigator award from the Fonds de Recherche en Santé du Québec (FRSQ).

The VERITAS project, as related to the RECORD Study, is supported by the Institute for Public Health Research (IReSP, Institut de Recherche en Santé Publique); the National Institute for Prevention and Health Education (INPES, Institut National de Prévention et d'Education pour la Santé) (Prevention Program 2007; 2010-2011 fınancial support; and 2011-2013 fınancial support); the National Institute of Public Health Surveillance (InVS, Institut de Veille Sanitaire) (Territory and Health Program); the French Ministries of Research and Health (Epidemiologic Cohorts Grant 2008); the National Health Insurance Offıce for Salaried Workers (CNAM-TS, Caisse Nationale d'Assurance Maladie des Travailleurs Salariés); the Ile-de-France Regional Health Agency (ARS, Agence Régionale de Santé d'Île-de-France); the City of Paris (Ville de Paris); the Ile-de-France Regional Council (Conseil Régional d'Île-de-France, DIM SEnT and CODDIM); and the Ile-de-France Youth, Sports, and Social Cohesion Regional Direction (DRJSCS, Direction Régionale de la Jeunesse et des Sports et de la Cohésion Sociale). The VERITAS project is also supported by the Canadian Institutes for Health Research (CIHR No. TOO-105427). This work was supported by the Canadian Institute of Health Research (Grant # 213338). Ruben Brondeel is supported by a grant from the EHESP School of Public Health (Ecole des Hautes Etudes en Santé Publique). Yan Kestens is supported by an investigator award from the Fonds de Recherche en Santé du Québec (FRSQ).

The authors are thankful to Julie Meline and Antoine Lewin for their useful advices.

List of tables

Table I. Means and standard deviations of environmental exposures in the street-network residential buffer, the perceived residential neighborhood, and in the activity spaces by urbanicity (n=4383) Table II. Differences in environmental exposures between the street-network residential buffer, the perceived residential neighborhood, and the activity space by urbanicity, using paired