Marc-Michel Rohé

Maxime Sermesant

Xavier Pennec

Barycentric Subspace Analysis: a new Symmetric Group-wise Paradigm for Cardiac Motion Tracking

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

Introduction

Understanding and analyzing the cardiac motion pattern in a patient is an important task in many clinical applications. It can give insight into a pathology, by evaluating for example how the cardiac function is affected by a cardiovascular disease and if a therapy is needed or not. On top of traditional simple parameters such as the ejection fraction (EF), it can also be used to compute more complex parameters -such as strains in different directions -giving deeper insight to the efficiency of the heart motion and function. The cardiac motion is usually studied by finding correspondences -the registration step -between each of the frame of the sequence and the first frame corresponding to the enddiastole (ED) image, yielding a dense displacement field that tracks the motion of the myocardium. Taking the ED image as a reference is natural as it is the starting point of the contraction of the heart which is the most important phase in evaluating the efficiency of the cardiac function but this specific choice can lead to important biases in quantifying the motion especialy at end-systole (ES) where the deformations to be evaluated are large [START_REF] Tobon-Gomez | Benchmarking framework for myocardial tracking and deformation algorithms: an open access database[END_REF].

In this paper, we propose a novel approach to study cardiac motion. Instead of taking an unique image as the reference to evaluate the motion, we build affine subspaces on the manifold of deformations encoding cardiac motion. There are different ways to extend the concept of principal affine spaces from an Euclidian space to something defined on manifolds. The simplest generalization is tangeant PCA, where a covariance matrix is build on the tangeant space of the Karcher or Frechet mean. In Principle Geodesic Analysys (PGA) [START_REF] Fletcher | Principal geodesic analysis for the study of nonlinear statistics of shape[END_REF], subspaces are spanned by the geodesics going through a point and the tangent vector is restricted to belong to a linear space of the tangent space. In this paper, we use more general type of family of subspaces on manifolds called Barycentric Subspaces which were first introduced in [START_REF] Pennec | Barycentric Subspaces and Affine Spans in Manifolds[END_REF]. With respect to the method previously mentioned, it has the benefit not to be controlled by the central value. This gives a more consistent framework to study data in the case the underlying distribution is either multimodal or simply not sufficiently centered.

In the context of deformation analysis in medical imaging, the points of the manifold corresponds to 3D images whereas the geodesic are deformations mapping two images together. Optimal paths (geodesics) are represented by the initial velocity field at the geodesic path resulting from the registration of images. In the first part of this article, we define the barycentric subspaces of manifold and introduce the way to compute the barycentric coefficients and the projection of an image on a Barycentric Subspace of dimension k based on k + 1 images. Instead of performing registration with respect to a single template, we build a subspace based on multiple references images and take advantage of the information of group-wise registration [START_REF] Yigitsoy | Temporal groupwise registration for motion modeling[END_REF], by building a Barycentric Template of dimension 2 parametrized by the barycentric coefficients. Experiments are conducted on sequences of healthy and pathological patients and show that the barycentric coefficients of both populations present significant differences and two clear clusters appear. Then, we improve the registration of cardiac motion by relaxing the regularization within the 2-dimensional barycentric template representing a cardiac sequence. Finally, we further improve the methodology by deriving a formula leading to symmetric and transitive registration.

Barycentric Subspaces in Deformation Manifolds

In this section, we introduce barycentric subspaces following the notation described in [START_REF] Pennec | Barycentric Subspaces and Affine Spans in Manifolds[END_REF]. In order to adapt the framework from Riemaniann Manifolds to the context of computational anatomy (image deformation analysis), we follows the framework of [START_REF] Joshi | Unbiased diffeomorphic atlas construction for computational anatomy[END_REF]. Working in the space of images M, we define I as a point of this space, which can be for example an image of a cardiac sequence and we identify paths to deformations. In the following, we will use (k + 1) points R j , the references images, on this Manifold as well as (k + 1) coefficients λ j the barycentric coefficients. The Barycentric Subspace of dimension k spanned Image Projection Fig. 1: Barycentric subspace of dimension 2 built from 3 references images (R 1 , R 2 , R 3 ). Î is the projection of the image I within the barycentric subspace such that v 2 is minimum under the conditions j λ j vj = 0 and v + vj = v j . by these points is then defined as the set of points (images) Î in M such that:

k+1 j=1 λ j --→ ÎR j = 0, where --→ ÎR j = log Î (R j )
is the smallest velocity field that registers I to R j . Contrary to the Riemaniann setting where we would have exactly R j = exp Î ( --→ ÎR j ), we obtain through registration an inexact matching that approximates the log vector. This is the tangeant vector of the geodesic shooting I to R j . In the following, we will place ourselves in stationary velocity fields (SVF) framework [START_REF] Vercauteren | Symmetric log-domain diffeomorphic registration: A demons-based approach[END_REF] which gives a simple and yet effective way to parametrize smooth deformations along geodesics using one-parameter sub-group. In this case, the tangent vector --→ ÎR j will be parametrized by the SVF vj and the condition simply becomes k+1 j=1 λ j vj = 0. The notation are summed up in Fig. 1.

Projection on Barycentric Subspace

Having defined the barycentric subspace spanned by a set of k + 1 references R j , we are looking to find the projection Î of any image I in M on this subspace together with the coefficient λ j representing the coordinates of Î within the barycentric template. The projection Î of I is the closest point to I that belongs to the barycentric subspace. We define the SVF v which parametrizes the projection of I such that -→ ÎI = v as well as the SVFs (v i ) i=1,...,k+1 such that --→ R i I = v i as shown in Fig. 1. The distance between I and Î is represented by the norm of the SVF v 2 . As seen previously, the constraint that Î belongs to the barycentric subspace can be written as j λ j vj = 0. Using the Baker-Campbell-Hausdorff (BCH) [START_REF] Vercauteren | Symmetric log-domain diffeomorphic registration: A demons-based approach[END_REF] formula, we get a first order development of v i = v + vi . The problem can now be written as:

min v v 2 , subject to i λ i (v i -v) = 0, i λ i = 0
whose set of solutions is:

λ = αA -1 1, α ∈ R * , A = v i |v j i,j .
With the additional constraint j λ j = 1, we get the unique solution by normalizing the λ. Finally, given a set of N images I n , we can define the distance of a barycentric subspace spanned by the references (R j ) j=1,...,M to the set of images (I n ) n=1,...,N as the sum of the error over all the residual projection vector vn such that:

E((R j ) j=1,...,M ) = n=1,...,N vn 2 , (1)
and find the closest subspace with respect to the cardiac sequence by minimizing this distance over all set of references.

Cardiac Motion Signature for Group-Wise Analysis of Dynamics

We applied the previously defined methodology to compare the cardiac motion signature of two different populations. The first group consists of 15 healthy subjects from the STACOM 2011 cardiac motion tracking challenge dataset [START_REF] Tobon-Gomez | Benchmarking framework for myocardial tracking and deformation algorithms: an open access database[END_REF]: an openly available data-set, and the second group is made of 10 Tetralogy of Fallot (ToF) patients. Short axis cine MRI sequences were acquired with T = 15 to 30 frames. The methodology described was applied by projecting each of the T frame of the cardiac motion to a barycentric subspace of dimension 2 spanned by 3 references. This set of 3 references is chosen by building the optimal barycentric subspace as induced by the distance defined in Eq. 1. Significant differences in the frame for the optimal references can be seen between the two populations (Fig. 2, left). In particular, the second reference -corresponding the the end-systole -is significantly higher for the ToF patients showing that this population has on average longer systolic contraction. Then we project the whole cardiac motion on the barycentric subspace made of these three references to compute the barycentric coefficients (Fig. 2, right). We see significant differences between both group of curves, especially in the region λ 1 < 0 showing that this signature of the motion is encoding relevant features and classical machine learning algorithms can separate the two populations using this representation.

Registration Using Barycentric Subspaces

In this section, we show how the use of barycentric subspaces as a prior on the cardiac motion can improve the registration by relaxing the regularization. Most of the registration methods rely on a trade-off between a fidelity-to-data term -capturing how well the registration matches the intensities of the voxels of the images -a smoothness regularization term -encoding our prior information about the regularity of the deformations we are looking. It is standard practice in registration algorithms to consider slowly varying deformation as our prior knowledge of the transformation (either by constraining the deformation to be within a small subspace of all diffeomorphisms or by penalizing large deformations). While this methodology works well to find small deformations, the regularization often leads to an underestimation of the large deformations as the one happening between the ED and ES frame. To overcome this drawback, solutions usually rely on performing the registration in a group-wise manner: a group of images are considered simultaneously and an additional criteria is added to ensure temporal-consistency [7] [8]. In this paper, we propose to use the barycentric template defined by 3 frames of the sequence as an additional prior on the transformations by considering that only the distance to the closest image within this barycentric subspace should be minimized. In the regularization step, we no longer consider the whole velocity field v but we run the regularization only with respect to v which encodes the distance of the current image to the closest image within the barycentric template representing the cardiac motion.

Barycentric Log-Demons Algorithm

We apply this methodology to the Symmetric Log-Domain Diffeomorphic Demons algorithm [START_REF] Vercauteren | Symmetric log-domain diffeomorphic registration: A demons-based approach[END_REF] which successively updates the velocity field to match the data, then smooths the velocity field with a gaussian filter. Instead of performing the regularization on the complete velocity field (the "standard" method), we decompose the velocity field v i as the sum of vi mapping the reference R i to the projection Î inside the template and the residual velocity field v of the projection (see Fig. 1) and we regularize only the residual v with the gaussian filter. The barycentric template is therefore used as a prior on the cardiac motion for which we do not perform regularization. The method was evaluated using a synthetic time serie of T = 30 cardiac image frames [START_REF] Prakosa | Generation of Synthetic but Visually Realistic Time Series of Cardiac Images Combining a Biophysical Model and Clinical Images[END_REF], so that we have ground truth meshes along the sequence allowing us to estimate the accuracy of the registration. First, we find the optimal references by minimizing the energy in Eq. 1 giving us the frames 1, 11 and 21 which will be the three references spanning the barycentric subspace. Then we register each frame i of the sequence using the method described above to get the deformations from each of the three references to the current images. We deform the ground truth meshes at the references frames with these deformations and compare the results with classical registration. As can be seen in Fig. 3, barycentric registration performs better at catching the end-systolic deformation with the contour of the warped mesh at end-systole matching better the ground truth. the estimation of the ejection fraction from the volume curve is also improved, going from 32% with the standard method to 38%, closer to the ground truth (43%), reducing the estimation error by half. Finally, the average point-to-point error for both methods shows (in Fig. 4, right) that, while barycentric registration has largest error for small deformations close to each reference, it has around 30% smaller error for largest deformations as between ED and ES.

Towards Symmetric Transitive Registration

In this last section, we quickly introduce a way to derive approximately consistent transitive (at the first-order of the BCH approximation) registration from the barycentric SVFs computed in the previous section. Symmetry and transitivity are two important properties for registration methods to improve robustness and reduce the unpredictability of the results [START_REF] Škrinjar | Symmetric and transitive registration of image sequences[END_REF]. A registration method is said to be symmetric if it associates two points regardless of the order of images that are registered together (in the SVF setting it is equivalent to v j i = -v i j ). Transitivity requires that the deformation given by the registration between two images should be equal whether it is done directly or by the composition of the result of the registration with an intermediate image (in the SVF setting it can be stated as

v j i = BCH(v k i , v j k ) v k i + v j k
with the BCH at the first order). Most registration methods fail to be transitive due to the accumulation of the registration errors at each step of the registration. Using Barycentric Subspaces as a basis for the registration at each step, we define the symmetric registration using the following formula which is schematically represented in Fig. 4:

W t s = vt -vs + 1 2 i (λ i s vt i -λ i t vs i ). (2)
In this formula, the first two SVFs on the left represent the residual transformations from the barycentric subspace to the two time points, and the sum on the right is a symmetric estimation of the SVF Ŵ s t within the barycentric subspace by going through each reference image forward and backward. This formula defines registrations that are both symmetric and transitive up to higher orders of the BCH in the compositions. It can be used for frame-to-frame as well as for frame-to-reference registration. In the former case, setting the reference to the first frame (s = 1 in the above formula) leads to improved results as shown in Fig. 4: the maximum error over the sequence is reduced by approx. 10% with respect to barycentric registration.

Conclusion

A new symmetric group-wise paradigm to study cardiac motion was proposed. Our approach relies on building subspaces as the reference for registration instead of choosing a specific arbitrary single image which can introduce bias. These subspaces represent the cardiac motion by meaningful parameters showing different clear patterns between two populations. Using these subspace as a prior, thereby relaxing the regularization on a 2-dimensional subspace, we achieve a better evaluation of the deformation between ED and ES frames and in particular we improve the estimation of the ejection fraction. Finally, the methodology can also be used to perform symmetric transitive registration, for better tracking along the sequence.

Fig. 2 :

 2 Fig. 2: (Left): Optimal references for the two group for each of the 3 references. (Right): Barycentric coefficient curves λ projected on the 2D-plan i λ i = 1.

Fig. 3 :

 3 Fig. 3: (Left): Image of heart at end-systole, the contour of the warped mesh of the initial frame 1 (ED) using barycentric (red), standard (blue) registrations are shown together with the ground truth (green). Using barycentric regularization, the registration is less constrained and we manage to get a more accurate contours for the end-systolic endocardium. (Middle): Error of the registration with respect to 3 reference images using the two methods, barycentric (dotted lines) and standard (plain lines). (Right): volume curves induced by the registration and comparison with the ground truth volume.

Fig. 4 :

 4 Fig. 4: (Left): Schematic representation of the symmetric multi-references barycentric registration in the case of a 1-D barycentric subspace spanned by 2 references. I t and I s are two frames of the sequence and Ît , Îs corresponds to the respective projection to the barycentric subspace. (Right): comparaison of the error between the standard registration (blue plain), the barycentric method presented in section 3.1 (blue dotted) and the symmetric-barycentric extension presented in section 3.2 (red dotted).

Ackowledgements The authors acknowledge the partial funding by the EU FP7funded project MD-Paedigree (Grant Agreement 600932)


Guido Petrucci
email: guido.petrucci@leesu.encp.fr

Fabrice Rodriguez

José-Frédéric Deroubaix

Bruno Tassin

Linking the management of urban watersheds with the impacts on the receiving water bodies: the use of flow duration curves

Keywords: Combined sewer overflows, Erosion, flow-duration curves, Model calibration, Modeling, Receiving water body, Uban hydrology

INTRODUCTION

A growing concern for urban stormwater management is related to its impact on the receiving water bodies. Impacts can range from acute or chronic pollution to streams' erosion or lakes' eutrophication, with severe consequences on both the possible uses of water bodies (e.g. water supply, fishing, recreational activities) and their environmental value. (Walsh et al., 2005b) The necessity to account for these impacts when planning or managing urban stormwater systems is widely recognized by scientists, technicians and decision-makers. However, this recognition is much more theoretical than practical: in current practices of urban hydrological studies, receiving water bodies are often treated as a secondary point and superficially, if they are treated at all. We can illustrate this point using the example of stormwater source control. Source control (also called Low Impact Development or Water Sensitive Urban Design) mainly consists in the implementation of small-scale stormwater facilities (often called Best Management Practices, or BMPs) over an urban catchment's area, upstream of the drainage system. One main principle of source control is to mimic the natural (i.e. pre-development) behaviour of the catchment, thus minimising the impacts of urbanization on the downstream environment (Booth andJackson, 1997, Roesner et al., 2001). This strategy is mainly implemented through source control regulations, demanding to include BMPs in any new urban development project. [START_REF] Balascio | A survey of storm-water management water quality regulations in four Mid-Atlantic States[END_REF] A recent study on source control regulations [START_REF] Petrucci | La diffusion du contrôle à la source des eaux pluviales urbaines. Confrontation des pratiques à la rationalité hydrologique[END_REF][START_REF] Petrucci | Do stormwater source control policies deliver the right hydrologic outcomes ?[END_REF], showed that the preservation of downstream water bodies is a commonly stated objective of local authorities adopting these regulations. Still, when technical studies are done to define the regulations, in most cases the only indicators considered are the peak flow-rates for extreme rainfall events (e.g. with return periods T ≥ 10 years). This attention on extreme flow-rates is the signal of the general persistence of a conventional approach, in which the only objective is to minimize sewer overflows and flooding, without regard to downstream water bodies (Petrucci, 2013).

On the basis of two observationsthe first about current modelling practices in urban stormwater management (section 1.1), the second about the link between catchment's urbanization, its hydrological behaviour and the impact on receiving water bodies (section 1.2)we propose, in this paper, a practical approach to integrate the receiving water bodies into current modelling practices, recurring to flow-duration curves (FDC). This proposal is described in section 2, and an example about source control regulations' analysis is given in section 3.

Current trends in modelling of urban stormwater systems and their impacts on receiving water bodies

Today, there are three main modelling trends in urban hydrology studies that, ordered by growing complexity and integration of receiving water bodies are: flow-rate modelling, quality modelling, integrated modelling.

Flow-rate modelling is the most current modelling approach: it consists in modelling the urban watershed and the drainage system, in order to simulate flow-rates in the sewer system. This modelling has a long history and tradition, and today is mainly represented by distributed, detailed models (e.g. SWMM 5, MIKE URBAN, InfoWorks, etc.). Many local authorities have already developed this kind of model for the urban areas they administer. Often they use them to determine interventions on the sewer system, or for more sophisticated uses like real-time control. The main limit in the current use of these models for integrating downstream water bodies, is the one we already noticed: these traditional models, although they have large possibilities, are often used in a traditional, narrow way. Often, they are calibrated on one or some important rain events, and they are applied just to simulate single events or short time series supposed to benchmark the functioning of the sewer system. It is worth to notice that, in general, nothing inside these models constrains to work on important rainevents: their structure is able to simulate low flow-rates as well as high, and current behaviours of the catchment as well as extreme ones. Moreover, most currently used models are able to simulate long, continuous time series instead of single rain events [START_REF] Elliott | A review of models for low impact urban stormwater drainage[END_REF].

The second modelling option is water quality modelling. This approach allows simulating the flow of pollutants from the urban area to the outlet. In principle, this type of modelling is extremely important to assess the impact on downstream water bodies, and its diffusion is an encouraging trend. Also because of legal constraints on stormwater quality, most flow-rate models now includes some waterquality simulation capability. However, because of a frequent lack of calibration data, high

measurement uncertainties, open questions on relevant physical and chemical processes, and in general a minor hindsight on this type of modelling compared to flow-rate modelling, water quality models have a smaller predictive capacity and reliability than flow-rate models. Uncertainties in predictions are amplified by a delicate modelling process, requiring significant efforts and know-how.

The third modelling option is integrated modelling. This term, defining the coupling of several models of different components of a system, is inclusive of many different modelling attempts in urban hydrology. The coupling between sewer systems and receiving water bodies is investigated, for example, by [START_REF] Silva | Coupling cyanobacteria dynamics and urban runoff modelling: an integrated approach for a tropical lake in Brazil[END_REF], modelling algal blooms in a urban lake as a consequence of changes in the upstream urban drainage. This type of modelling, today, is still the object of single case-specific researches, and cannot be considered as a common practice for stormwater management purposes.

In summary, the most current and solid approach is, today, flow-rate modelling. However, this approach is extremely focused on sewer overflows, and do not take into account downstream effects.

The development of other models to a level of easiness of use and reliability sufficient for a wide use by local authorities will probably take years. Because the purpose of this paper is to find a practical and viable solution to improve the accounting for impacts on the receiving water bodies, we will search it in the framework of flow-rate modelling approach, exploring their unexploited potential;

The impact of urbanization on downstream water bodies

The most evident effect of urbanization on catchment hydrology, recognized since several decades (e.g. [START_REF] Leopold | Hydrology for Urban Land Planning. A Guidebook on the Hydrologic Effects of Urban Land Use[END_REF], is an increase in peak flow-rates. This phenomenon was particularly noticed for large rain events, when downstream floods or rapid stream erosion were observed. The efforts in stormwater management were thus directed, for a long time, to control this kind of high-flow events. Typical measures in the US (Roesner et al.,2001, Balascio and[START_REF] Balascio | A survey of storm-water management water quality regulations in four Mid-Atlantic States[END_REF] or in the UK [START_REF] Faulkner | The control of surface water runoff from new development -UK national 'policy' in need of review?[END_REF] address rain events with return periods higher than 1 or 2 years. In France [START_REF] Petrucci | Do stormwater source control policies deliver the right hydrologic outcomes ?[END_REF] most stormwater systems (sewers, but also BMPs and large retention basins) are dimensioned for a return period of 10 years. This long-lasting practice of focusing only on relatively uncommon events is what we still find in a large majority of urban hydrological studies.

Even if high-flow uncommon events surely have an impact on the receiving water bodies, since the 90's a consistent literature (summarized by Walsh et al., 2005b) started to reconsider their importance, in comparison with more current events. In terms of channel erosion, for example, it has been shown that erosive flow-rates are small enough to be exceeded even by small and frequent rain events [START_REF] Hunt | Relating stormwater control measure (SCM) design standards to stream erosion in piedmont North Carolina: case studies in Raleigh and Chapel Hill, North Carolina[END_REF]. The latter, thus, are likely to have a stronger effect, on the long term, than large uncommon events [START_REF] Booth | Urbanization of aquatic systems: degradation thresholds, stormwater detection, and the limits of mitigation[END_REF]. The same analysis on the increased frequency of exceedance of a "disturbance threshold" because of small rain events is suggested for the biological conditions of streams [START_REF] Booth | Reviving urban streams: land use, hydrology, biology, and human behavior[END_REF], Roy et al., 2005). In terms of water quality, two effects should be mentioned: the first, described by Walsh et al. (2005a) is the production of runoff, not occurring in natural catchments, for rain events of a few millimetres. This frequent runoff, even if hydraulically negligible, may constitute a relevant and in some cases chronic intake of pollutants (and heat). The second effect is the increase in combined sewer overflows.

In summary, there is a growing recognition of the importance of low and medium flow-rates. The behaviour of an urban catchment during extreme rain events is significant in terms of urban flooding, thus representing a point of view focused on what happens inside the urban area. If we are interested in what happens downstream, we should look at the current behaviour of the urban catchment.

CONSTRUCTION AND USE OF FLOW DURATION CURVES

As a consequence of the growing interest for current urban catchments behaviour, some authors suggested hydrologic metrics linked to downstream effects. [START_REF] Booth | Reviving urban streams: land use, hydrology, biology, and human behavior[END_REF], for instance, suggested using the fraction of days, on an annual basis, when the daily mean discharge exceed the annual mean discharge (T Qmean ). This indicator can distinguish catchments that have more "flashy" response (low T Qmean ) from other with gradually varying flow regimes (high T Qmean ). However, the information given by so specific indicators is difficult to generalize to other geographic, climatic and urban conditions from that where the indicators were developed and tested. That is why several researchers (e.g. [START_REF] Fennessey | Changes in runoff due to stormwater management pond regulations[END_REF][START_REF] Roesner | Physical effects of wet weather ows on aquatic habitats: present knowledge and research needs[END_REF][START_REF] Rohrer | Matching the critical portion of the flow duration curve to minimise changes in modelled excess shear[END_REF] used a more general way to characterise catchments' current behaviours, using Flow Duration Curves (FDC).

A FDC represents the fraction of time during which a given level of flow-rate is equalled or exceeded at a point of the drainage network [START_REF] Vogel | Flow-Duration Curves. I: new interpretation and confidence intervals[END_REF]. This kind of representation has been largely employed in water management, because of its capacity to represent a huge quantity of hydrological information on a single view (Vogel and Fennessey;[START_REF] Vogel | Flow-Duration Curves. II: a review of applications in water resources planning[END_REF]. Considering the observations made on section 1.2, we can compare the effect of urbanization as observed on a hypothetic hydrograph plot and on a FDC (figure 1). In the hydrograph (left) one can spot the increase in peak flow-rate, in runoff volume (area under the curve) and in the response "flashiness" (shorter ascending and descending limbs of the hydrograph). On the FDC (right) it is possible to observe:

 the increase in high flow-rates. The intersection of the FDC with the y-axis represents the highest flow-rate measured during the observation period (or a similar statistic, according to the procedure followed for the construction of the curvesee next section);  the increase in the runoff frequency. The intersection with the x-axis represents the fraction of time the catchment produces some runoff;

 the generalized increase of the frequency of exceedance for each flow-rate;

 indirectly, the increase in runoff volume, because it is proportional to the area under the FDC.

In substance, together with some elements on extreme behaviours of the catchment, this kind of representation provides much more information about the impacts on the downstream water bodies.

Construction of the flow duration curves

Starting from a flow-rate time series, the simplest way of constructing a FDC is to order the measurements by decreasing flow-rates, and to plot the result in frequency-flow-rate axes. More formally, starting from a series of n ordered flow-rates q (i) , where i = 1,…,n, the value of the FDC Q p , for the frequency p, is:

Q p = q (i) if i = [(n+1)p] Q p = q (i+1) if i < [(n+1)p]
where [(n+1)p] is the integer part of (n+1)p. This formalism, as well as more sophisticated and statistically robust method to calculate the FDC can be found in [START_REF] Vogel | Flow-Duration Curves. I: new interpretation and confidence intervals[END_REF].

The work of these authors on FDC is particularly interesting because it addresses a major inconvenient of this instrument: the strong dependence on the period used for calculation. In particular, the extremes of the curve are highly dependent on the extreme values of the time series, and they are poorly reliable. The solution suggested by [START_REF] Vogel | Flow-Duration Curves. I: new interpretation and confidence intervals[END_REF] is, for time series spanning over several years, to calculate several annual FDCs instead of a unique FDC for the whole recording period. Starting from the annual FDCs, they suggest calculating a median annual FDC, together with its confidence intervals. This method generates both more reliable FDCs and an estimation of the inter-annual variability of the hydrological regime of the catchment.

Another point, useful for the following discussion, is that, starting from a time series recorded at a given time-step (e.g. 5 minutes) it is straightforward to construct time series with time-steps multiple of the original one (e.g. 10 minutes, 1 hour, 1 day), by averaging subsequent records. Thus, starting from a time series at a given time-step, it is possible to obtain FDCs at different, longer, time-steps. Starting from a time series, it is possible also to construct curves other than FDCs but with similar construction procedures and meaning: for example, [START_REF] Fennessey | Changes in runoff due to stormwater management pond regulations[END_REF] build a quasi-FDC starting from a time series of the peak flow-rates for a series of rain events.

The use and the interpretation of FDC in urban hydrology

As observed in section 1.1, most currently used flow-rate models can perform continuous simulations over long periods of time. Thus, for any scenario of stormwater management that can be described by these models, it is possible to obtain long-term simulated time series and, consequently, the corresponding FDC. Scenarios can include new infrastructures (e.g. reservoirs), new operation rules (e.g. activation rules for pumps), new regulations (e.g. source control) and new urban developments.

FDCs can then be used to compare alternative scenarios. This can be done directly on the FDCs themselves (e.g. [START_REF] Fennessey | Changes in runoff due to stormwater management pond regulations[END_REF]Konrad and Burges, 2001), providing in a single view a comparison of the changes of the hydrologic regime consequent to the adoption of each alternative.

When the comparison involves many alternatives, and/or some case-specific objective is set and quantified, it is possible to define indicators on the FDC. To facilitate comparisons, one can use generic indicators (Petrucci, 2012) like, for example, the flow-rates exceeded for given fractions of the year (Q 0.1% , Q 1% and Q 10% in figure 2), or the intersections between FDCs and the axes (f 0 in figure 2). The most interesting application of FDC indicators, however, lies in the possibility to define parameters linked to receiving water status on a specific case-study. The indicator T Qmean proposed by [START_REF] Booth | Reviving urban streams: land use, hydrology, biology, and human behavior[END_REF] as explanatory of the downstream biological status can easily be computed from FDCs. If an "erosive threshold" can be estimated [START_REF] Hunt | Relating stormwater control measure (SCM) design standards to stream erosion in piedmont North Carolina: case studies in Raleigh and Chapel Hill, North Carolina[END_REF], its frequency of exceedance can be compared for different scenarios. In the same way, the frequency (and the volume, through some supplementary consideration) of combined sewer overflows can be estimated if the overflow threshold is known. Also the f 0 indicator defined above, describing the annual frequency of runoff generated by the catchment, can be a significant indicator of water quality and biological disturbance according to Walsh et al. (2005a,b).

As anticipated, it is possible to adapt the FDC time-step to the time-scales of the system studied. Models of urban stormwater systems often use considerably short time-steps (5'-10') in order to finely describe the hydraulic system. A similar time-step can be appropriate for computing, for example, combined sewer overflows, but a coarser time-step (hourly, daily) can be more adapted to describe impacts on systems (e.g. urban lakes) having longer response-time.

Required changes in current modeling practices

If, in general, each scenario that can be tested in terms of peak flow-rate can be tested in terms of FDC, three main requirements have to be satisfied.

The first is the availability of local rainfall data over long periods: peak flow-rate analyses could be based on a few (or even just one) real or synthetic rainfall events, while for long term simulations, representative input data is necessary. However, this requirement can be easily fulfilled thanks to the raingage networks that are in place since many years in most countries, particularly for urban areas.

The second is the calculation capability necessary to run long term simulations. Still, with the exception of very large and complex urban catchments, requiring specific solutions, present calculation resources can in most cases run hydrologic simulations in reasonable computation times.

The third and, by our point of view, more problematic requirement is that the model should be reliable when simulating catchment's current behaviours. Today this is not always the case, because, even if flow-rate models are able to simulate the whole hydrologic regime of a catchment, in practice they are often calibrated on a few high flow-rate events for which measurements are available. This procedure focused on peak flow-rate does not guarantee the reliability of the model (i) for low flow-rate events and (ii) for the parts of the hydrologic response different from the peak (i.e. ascending and descending limbs of the hydrograph). Now, the reliability of the model in these terms is a primary condition for the reliability of the FDCs obtained.

The solution to this reliability issue demands the probably most important evolution in current modelling practices. In facts, it requires to calibrate the model on low flows as well as on high flows, and considering the whole hydrograph shape and not only the peak position and amplitude. Further, an event-based calibration is not sufficient: time series of input (rainfall) and output (flow-rate) are necessary. While rain data are easily available, flow-rate time series are less common. Further, calibration of flow-rate models is complex because typical urban catchments' models can involve hundreds to thousands of parameters, often difficult to estimate a priori. To calibrate just on some peaks, manual calibration of a few parameters can often be satisfactory, but to fit the model for the whole catchment's response, a better procedure should be find. Technical analyses of urban stormwater management should start to include automatic calibration methods, quite diffused in hydrologic research, but barely applied in practice.

AN EXAMPLE OF THE APPLICATION OF FDC IN URBAN HYDROLOGY

Methodology

We propose the application of FDC analysis to the comparison of different source control regulation scenarios. This example is intended to show the better insight given by FDCs on the catchments' behavior.

We considered the urban catchment of "les Gohards", in Nantes, France (figure 3), already studied by [START_REF] Rodriguez | Surface runoff in urban catchments: morphological identification of unit hydrographs from urban databanks[END_REF]. This catchment (178 ha) is covered by a mixed land-use (residential, commercial, industrial, agricultural), giving an impervious cover of about 0.38. The sewer system is separated (14.8 km); its average slope is 0.79%. Available data include 5 years of flow-rate measurements at the outfall (1999)(2000)(2001)(2002)(2003), and 10 years of rainfall measurements at two raingages inside the catchment (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008). Both time series have a 5' time-step.

The model of the catchment is realized in SWMM 5 [START_REF] Rossman | Storm Water Management Model (SWMM version 5.0) user's manual[END_REF], using typical methods for subcatchments' delineation and parameters' estimation (e.g. [START_REF] Gironàs | Storm water management model applications manual[END_REF]. The only particular feature of the modeling setup is that we distinguished, in each "real" subcatchment, roof, road and green areas, in order to prepare the simulation of source control regulation scenarios. Details on this procedure are given in [START_REF] Petrucci | Do stormwater source control policies deliver the right hydrologic outcomes ?[END_REF] After setup, the model was calibrated and validated. As for the model setup, all choices in the calibration/validation procedure were a compromise between accuracy and simplicity: the purpose was to reach our aim using the most current options in hydrological literature, in order to minimize the changes from current modeling practices.

For calibration, we used only one month of the available rainfall/runoff data, to keep the computational requirements low. The period was selected for data quality (no gaps) and for the variety of rainfall events occurred. Calibration parameters were nine global parameters (roughness of pipes and surfaces, infiltration, initial losses, etc.) and one shape parameter (length of the overland flow path) for each subcatchment, for a total of 101 parameters. To cope with so many parameters, calibration was performed using a genetic algorithm, an automatic optimization algorithm largely applied in hydrology in the last 20 years [START_REF] Savic | Evolutionary computing in hydrological sciences[END_REF]. Optimization consisted in maximizing the Nash criterion [START_REF] Nash | River flow forecasting through conceptual models part I: A discussion of principles[END_REF].

Validation was performed calculating the Nash criterion for each season (4 months) of the remaining data, in order to verify that the choice of a single month, in autumn, for calibration, was not reducing its performances in other periods of the year.

Scenarios

The scenarios compared are of two types: specific flow-rate regulations, demanding to store stormwater and to release it at a limited specific flow-rate q* (l/s/ha), and volume regulations, demanding to store and infiltrate a given volume i* (mm) of stormwater. For more details on this kind of regulations and their modeling, see [START_REF] Petrucci | Do stormwater source control policies deliver the right hydrologic outcomes ?[END_REF]. In this example, we consider source control regulations applied systematically and homogeneously all over the catchment. However, the same procedure can be applied to more realistic scenarios, with partial or non homogeneous applications of source control. Specific flow-rate regulations were modeled by installing a reservoir downstream of the roof and road areas of each subcatchment. Each reservoir has a volume and an outfall sized as a function of q*, according to French "rainfall-based" sizing procedure. Volume regulations are modeled as a pervious filter strips downstream of each impervious area (figure 4). The storage of the strip is sized as a function of i*.

Each scenario was simulated on a 10 years period with a 5' time-step. We also simulated a synthetic rainfall having a return period of 10 years (triangular, 1-hour duration), in order to compare a classical peak flow-rate approach to FDCs.

Results and discussion

Calibration and validation

The calibration procedure provided a high value of the Nash criterion (0.91), meaning a good accuracy of the model. Visual verification was also satisfactory (figure 5). In validation, we calculated the Nash criterion for nine seasons with available data, obtaining values ranging between 0.72 and 0.89 (median: 0.85), without evident seasonal biases.

Scenarios' simulation

In figure 6 are plotted the median annual FDCs for, respectively, specific flow-rate and volume regulations. The reference value (black line) is the FDC calculated by the model with no scenario applied. The two sets of FDCs show immediately a significant difference of behaviors between the two types of source control regulations: while specific flow-rate regulations reduce the frequencies of high flow-rates, but increase those of small ones, volume regulations systematically reduce frequencies. When specific flow-rate regulations become stricter (i.e. smaller q*), highest flow-rates progressively disappear, but low flow-rates are increasingly common, exceeding the frequencies of the reference case. The effect of specific flow-rate regulations can be summarized by a flattening of the hydrological behavior of the catchment. The atypical behavior of the q*=0,5 l/s/ha curve is because of the sizing procedure, not adapted for long reservoir's emptying times: with a so low outlet flow, complete emptying can require more than one week, causing spills for subsequent rain events. A similar sizing problem cannot be spotted by single-event simulations, but easily appears with long term simulations.

If the comparison of FDCs allows identifying global trends, FDC-based indicators can help decisionmaking. In figure 7 are plotted the indicators Q 10years , corresponding to the peak flow-rate for a 10years design rainfall (26,8 mm in 1 hour) and Q 10% , representing the flow-rate exceeded 10% time of the year (i.e. 36,5 days).

The first graph, representing a classical indicator for stormwater management decision making shows a linear reduction of peak flow-rates (red line) increasing q*. This justifies a general trend, among local authorities, to progressively reduce the q* value in regulations: the stricter the rule, the smaller the flow-rates to be managed downstream, the less sewer overflows and other nuisances.

The second graph, issued from the frequency analysis of long term simulations, presents a different behavior of flow-rate regulations: the indicator has a significant increase for low values of q*. If there was a constraint on downstream flow-rates, for example an erosive threshold for the downstream creek (or the activation threshold for a combined sewer overflow) estimated at 1 l/s/ha (about 200 l/s), the graph could show that regulations ranging from q*=5 l/s/ha to q*=1 l/s/ha (quite common in France) will significantly worsen the creek's erosion. More generally, this example shows that protecting the environment downstream of a urban area can be an objective contrasting with the protection of the urban area alone. Volume regulations seem able to attain these two objectives simultaneously, but it is not possible to understand this point if only single-event, peak flow-rate based analyses are done. Taking into account downstream water bodies through FDCs can improve their protection and cast doubts on the current practices of source control regulations.

CONCLUSIONS

In this communication, we presented the interest of analyzing urban stormwater management by the mean of FDCs. We argued that this instrument can provide useful information on the impacts on downstream water bodies without requiring significant changes in the way urban water studies are done today. We think that FDCs represent a practical approach to improve the protection of water bodies.

Two final remarks: the first is that FDC analysis, as we showed in our example, is not alternative but complementary to classical peak flow-rate analyses on single events. The concern for water bodies' protection is growing, but the first purpose of urban stormwater systems remains the protection of people and properties from flooding and other nuisances. Decision-making in this domain is a search for a compromise between different objectives. FDC analysis, in this context, is just a tool to facilitate the search for a better compromise. The second remark is on the meaning of FDCs: if they actually provide much information on downstream impacts, this information has to be interpreted according to the specific context. A wise use of FDCs requires to understand which changes in the hydrological behavior are more harmful (or positive) to the specific downstream environment. Local researches on water-bodies are necessary to better manage them.

Figure 2 -

 2 Figure 2 -Examples of indicators defined on an annual median FDC. Blue and green lines represent the confidence intervals at 90% (15 years of simulation at 5' time-step). Source: Petrucci (2012).

Figure 3 -

 3 Figure 3 -The Gohards catchment. Source: Petrucci, 2012.

Figure 4 -

 4 Figure 4 -Scheme used for volume regulations' modeling. Source: Petrucci et al., 2012.

Figure 5 -

 5 Figure 5 -Examples of results from calibration (left) and validation (right).

Figure 6 -

 6 Figure 6 -Median annual FDCs for flow-rate (left) and volume (right) regulations.

  Figure 7 -Q 10% (left) and Q 10years (right) indicators.

ACKNOWLEDGMENTS

We thank Nantes Métropole (Direction des Informations Géographiques) for providing the GIS data about the catchment.


Lomig Hamon
email: lomig.hamon@mines-nantes.fr

Clément Garénaux

Pascaline Pré

Kinetic separation

1. D eff /R 2 are obtained from the slope of the desorption curve in the long time region (equation 8). 2. Determination of the kinetic regime for different gas velocities (from 10 to 100 mL min -1 ) and for temperature from 303 K up to 393 K

The kinetic regime is reached for gas velocities greater than 50 mL min -1 : <50 mL min -1 : both thermodynamics and kinetics controls >50 mL min -1 : constant diffusivity independent of the gas flow velocity, i.e., kinetics control

𝛽 𝑛 𝑐𝑜𝑡𝛽 𝑛 + 𝐿 -1 = 0 (6) 𝑐 𝑐 0 = 2𝐿 𝑒𝑥𝑝 -𝛽 𝑛 2 𝐷 𝑒𝑓𝑓 𝑡/𝑅 2 𝛽 𝑛 2 +𝐿 𝐿-1 ∞ 𝑛=1 (5) 𝜕𝑞 𝜕𝑡 = 𝐷 𝑒𝑓𝑓 𝜕 2 𝑞 𝜕𝑟 2 + 2 𝑟 𝜕𝑞 𝜕𝑟 (1)

Pure CO 2 desorption experiments

Carried out on a home-made apparatus similar to the one developed by Eić and Ruthven [1] with a constant partial pressure of 5 kPa of CO 2 in the He matrix (total pressure of 1 atm). 45.5 mg of CMS has been used for these experiments.

Characterization of the CMS

CMS is provided by CarboTech. Its micropore volume is obtained by CO 2 adsorption isotherm measurement at 273 K: 0.22 cm 3 g -1 for pore diameters smaller than 0.7 nm. The SEM analysis of the microstructure shows the different pore morphologies [5].

From uptake curves, the CO 2 adsorption is a very fast phenomenon whereas the adsorption of CH 4 is slow due to the exclusion of CH 4 molecules. From Arrhenius plot, there is no evidence of the effect of the CH 4 on the effective diffusivities whatever the temperature.

From the plot of 1/L' as a function of 1/Q, the decrease of the slope when CH 4 concentration increases corresponds to a decrease of the intraparticle CO 2 diffusivity. However the ratio of the intraparticle diffusivity over the external resistance mass transfer (k) explained by the y-intercept keeps constant whatever the CH 4 concentration. Affinity constant (bar -1 ) K (mmol g -1 bar -

  10 th International Symposium on the Characterization of Porous Solids COPS-X | Granada, 11-14 May, 2014 Zero Length Column (ZLC) theory ZLC experiment are carried out by preequilibrating a small sample of adsorbent with a steam at a low concentration of the adsorbable compound in He carrier, and then by desorbing the adsorbable compound in pure He steam Hypotheses •ZLC cell well mixed •Linear adsorption equilibrium between fluid and adsorbed phases •Constant diffusivity (do not depend on the internal adsorbed phase concentration) •Spherical adsorbent particles •Desorption controlled by intraparticle diffusion Mass balance: Initial conditions: Boundary conditions: Analytical solution: Where b n 2 is given by the roots of: And L is defined as: For large values of t, then only the first term of the equation (5) is significant, and for large values of L, then b 1  p : With surface resistance: Change of the boundary condition Expression of the apparent L':

2 -

 2 CH 4 mixtures desorption experimentsConstant partial pressure of CO 2 of 5 kPa and variation of CH 4 partial pressure to give CO 2 -CH 4 mixtures 67-33, 50-50, and 33-67. The desorption of CH 4 is too slow to be analyzed: the MS detector does not allow to measure it.