Kian Rhabarnia

Benjamin P Brown

Mike M Clark

Elliot J Kaplan

Mark D Nornberg

Alex M Rasmus

Cary B Forest

Frank Jenko

Angelo Limone

Jean-Francois Pinton

Kian Rahbarnia
email: kian.rahbarnia@ipp.mpg.de

Nicholas Zane Taylor

Jean-François Pinton

Nicolas Plihon

Gautier Verhille

Direct observation of the turbulent emf and transport of magnetic field in a liquid sodium experiment

Keywords:

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

INTRODUCTION

Modeling astrophysical processes like stellar convection and accretion disk turbulence is accomplished only by assuming diffusivities for momentum, temperature, and magnetic fields far in excess of microphysical values. This assumption is one of the main challenges in the description of high Reynolds number flows and is a fundamental requirement for numerical modeling due to the many decades of separation between large-scale flows and dissipation scales. Molecular diffusivity, however, is irrelevant in these large systems compared with the effective mixing driven by turbulence in explaining the transport of momentum in accretion disks (Ji et al. 2006) and the transport of magnetic fields in the Earth and the Sun (Miesch & Toomre 2009;Glatzmaier & Roberts 1995).

Much effort at modeling turbulent transport has concerned the transport of scalar fields (e.g., temperature, concentration). Mixing length models provide a basic formulation of the problem (e.g., Böhm-Vitense 1958;Germano et al. 1991;[START_REF] Pope | Turbulent Flows[END_REF]. The transport of vector fields, like the magnetic fields that can dominate dynamics in astrophysical objects, is much more complicated. For instance, the topological properties of the flow are often essential for the description of elementary processes. Even in simple systems it is unclear whether the turbulent transport is adequately captured by parameterized treatments, e.g., if the transport is simply an enhanced diffusion or a combination of two or more underlying processes. Measurements of the turbulent electromotive force (emf) in low-beta plasma regimes (Ji & Prager 2002) and direct numerical simulations have studied the transport of magnetic fields and suggest that simple transport models might actually apply (Brandenburg & Subramanian 2005a), but, until now, only partial aspects have been confirmed experimentally (Reighard & Brown 2001;Stepanov et al. 2006;Spence et al. 2007;Frick et al. 2010).

Here we report the direct measurement of the threedimensional (3D) turbulent electromotive force in isotropic turbulence where hydrodynamic instabilities drive the fluctuations and the magnetic stress is weak. Our essential finding is that the turbulent motions enhance the magnetic diffusivity: they generate an emf in opposition to the local current density. The measured emf is consistent with the calculated resistivity from mean-field theory.

MAGNETOHYDRODYNAMIC TURBULENCE

Our present understanding of turbulence is still fragmentary. In particular the turbulent transport of vector quantities like momentum or magnetic fields is an area of active study. Turbulence is the natural state of all fluid or plasma systems that are driven sufficiently strongly. Turbulent flow patterns and dynamics are complex, and one of the key features of turbulent flows is that they lead to enhanced mixing. In hydrodynamic (non-magnetic) turbulence, mixing of momentum is the dominant process. Momentum is transported by nonlinear velocity interactions which are described by the Reynolds stress tensor (τ ≡v i v j , i.e., the correlation in the fluctuating velocity components). Measurements of τ have been essential in the evaluation of hydrodynamic turbulent transport models (Speziale 1991)-for instance, τ has been measured directly in model flows (Tao et al. 2002), in coastal ocean regions especially in the presence of surface waves (Rosman et al. 2008), and also has been measured in magnetically confined fusion plasmas (Hidalgo et al. 2000).

In magnetohydrodynamic (MHD) flows, the transport of magnetic flux is governed by the turbulent electromotive force (emf). Small-scale turbulent fluctuations in the velocity v induce magnetic field fluctuations b. They coherently interact to produce, on average, an emf E,

E ≡ v × b . (1)
It is this turbulent emf that we study in our experiment by measuring v and b directly. E affects the large-scale fields in the induction equation

∂ B ∂t =∇×( V × B + E) + η μ 0 ∇ 2 B , (2)
where • denote either a time average or a spatial average, and V , B are the mean components of the flow and magnetic field, v, b represent the fluctuating parts of flow and field, η is the resistivity, and μ 0 the vacuum permeability. In analogy with the turbulent eddy viscosity models developed for simple hydrodynamic turbulence, the so-called mean-field theory of MHD turbulence (Blackman & Field 2002;Brandenburg & Subramanian 2005b;R ädler & Stepanov 2006) has proposed closure models in which E is related to the large-scale magnetic field and its derivative. For isotropic, homogeneous, helical turbulence, E is expressed as

E = α B -βμ 0 J (3)
with α, β transport coefficients related to the statistical properties of the turbulent flow. This simple parameterization leads to very mathematically efficient models of astrophysical dynamos (Ruzmaikin et al. 1989). The α contribution is often attributed to the stretching and twisting of magnetic fields lines (Pétrélis et al. 2003). The β contribution corresponds to an enhanced magnetic diffusion, analogous to the eddy viscosity in hydrodynamic turbulence. In the idealized framework of homogeneous and isotropic fluctuations, β is a scalar quantity which can be expressed as β ≈ v 2 τ corr /3 ∼ṽℓ/3, where τ corr is the autocorrelation time of the velocity fluctuations and we have defined a correlation length ℓ =ṽτ corr . Introducing the magnetic Reynolds number Rm turb ≈ μ 0 vℓ/η of the fluctuations, the net magnetic resistivity is obtained from Equation (2)asη turb ∼ η(1 + Rm turb ).

Compared to laboratory experiments in astrophysical systems, Rm turb ≫ 1 is usually enormous, and very large magnetic resistivities are expected. When applied to dynamo generation in the context of mean-field models, this enhanced resistivity may inhibit the magnetic field generation. It may be balanced by the α term (here mainly related to the flow helicity) which corresponds to a current generation in the direction of the background, large-scale, magnetic field-it is controversial in very turbulent systems (Hughes & Cattaneo 2008;Käpyläetal.2009;Hughes et al. 2011;Blackman & Field 2002), in part due to a strong predicted quenching as dynamos transition from unmagnetized to magnetized states. Whatever the magnitude of the α-effect is, the existence and characterization of the β-effect and an enhanced resistivity η turb is essential as it greatly increases the transport of magnetic fields. It is addressed here using the liquid sodium Madison Dynamo Experiment (MDE; shown in Figure 1) and related numerical modeling.

THE MADISON DYNAMO EXPERIMENT AND DIAGNOSTICS

MDE studies the role of turbulent magnetohydrodynamic processes in the onset conditions for magnetic field growth. Several laboratory experiments have generated self-sustained dynamos in turbulent flows either strongly constrained in geometry (Gailitis et al. 2001) or using ferromagnetic boundaries (Verhille et al. 2010). In many natural astrophysical and geophysical systems, like our Sun (Miesch & Toomre 2009), other stars (Brown et al. 2010), or Earth's core (Kageyama et al. 2008), the underlying flows are unconstrained. Very high Reynolds numbers and strong shear in the flow of MDE lead to vigorous turbulence. Measurements using water in a similar apparatus show that, at smaller scales, the turbulence is locally isotropic and homogenous (Nornberg et al. 2006), while at large scale it has strong poloidal and toroidal circulation (Dudley & James 1989). The experiment is operated in a regime where the magnetic field plays a limited role in the fluid dynamics and any back-reaction of the magnetic field on the flow can be neglected (Nornberg et al. 2006). Within this kinematic regime the magnetic field can be considered as a passive vector transported by the turbulent flow. In the case of strong growing magnetic fields, the back-reaction can no longer be neglected and simple parameterization models, like Equation (3), no longer apply. Although the measurement technique for observing the turbulent emf, which is used for the present studies, is still valid, the interpretation of the results requires more complex models (Bhattacharjee & Yuan 1995). Nevertheless the kinematic approach covers most important basic features in many astrophysical systems (Brandenburg & Subramanian 2005a;Ruzmaikin et al. 1989). For the first time it is possible to directly observe all three components of the turbulent emf at a local position in the experiment. The probe used for direct measurements of the local turbulent emf (in the following paragraphs called the 3D emf probe) is based on a miniaturization (Noskov et al. 2009) of the well-known principles for velocity measurements in liquid metals by Ricou & Vives (1982). Compared to probes that have been used in other experiments (Noskov et al. 2009;Ricou & Vives 1982;von Weissenfluh 1985), our newly designed 3D emf probe, which will be described in detail in the following two paragraphs, combines a 3D velocity component measurement with a simultaneous measurement of all three magnetic components.

For each of the three directions of the velocity, radial r, latitudinal θ , and longitudinal φ, the potential difference between two silver-coated copper electrodes which hold a small NdFeB permanent magnet (size: 2 × 2 × 10 mm 3 , surface field: 0.2 T) in between, is measured simultaneously (Figure 2). Due to the strong local magnetic field of the permanent magnet B pm , which is two orders of magnitude higher than the background field, the potential difference Φ = Here f cs ≈ 80 kHz with d = 1 mm, which is a typical length for changes of the strong local magnetic field given by the size of the permanent magnets used. Compared to alternative methods for flow measurements in liquid metals (e.g., applying strong global magnetic fields, where the relationship between the measured potential and flow becomes uncertain, imposing a local magnetic field with a permanent magnet yields a direct approximation of f cs which, for the present work, is well above the relevant driving frequencies in the flow (Reuter et al. 2011). P 1 and P 2 are the positions of the two electrodes and their distance is l = 3 mm. As in the hot wire technique, the distance between the electrodes sets the spatial resolution of the measurement, which leads to a cutoff frequency (Bolonov et al. 1976)o ff co = 1.5F (R/l) (2/3) ≈ 0.14-1.4 kHz with the rotation frequency of motors F = 3-30 Hz and the radius of the sphere R = 0.5m .f co is still about 1-2 orders of magnitude higher than typical driving frequencies in the flow. The measured potential differences are amplified with an amplifier type AM502 and processed by 16-bit digitizers on PC-based data acquisition cards with a sample rate of 1 kHz per channel. The signal-to-noise ratio of the velocity measurements is S/N V ≈ 10 2 . The time-resolved signals are filtered by a standard second-order Butterworth filter at 15 Hz and the resulting mean velocities are averaged over the length of the electrode.

The tip of the 3D emf probe also holds a set of magnetic Hall sensors (4 cm away from the latitudinal and longitudinal electrodes; Figure 2), which measure the 3D mean and fluctuating parts of the magnetic field B meas = B + b. These Hall sensors are far enough from the probe tip to neglect the induced field due to the advection of the magnetic field of the small permanent magnet, but well within the magnetic diffusion length of about 10 cm (Nornberg et al. 2006). The signal-to-noise ratio of the magnetic measurements is S/N B ≈ 10 4 . Simultaneous measurements of v and b allow us to obtain the local turbulent emf directly.

In addition to the single 3D emf probe described above, magnetic fields B in MDE are measured by about 300 Hall sensors inside and on the surface of the experiment. Their positions can be seen in Figure 1(a). To determine the local current density J the derivatives of B are needed. In an axisymmetric spherical geometry it is common to convert the magnetic and velocity fields from a spatial domain to a spectral domain by decomposing them into vector spherical harmonics (Dudley & James 1989) represented by

B(r, θ , φ) = i ∇×∇×S i (r)Y m i ℓ i (θ,φ)r + ∇×T i (r)Y m i ℓ i (θ,φ)r. ( 4
)
B(r, θ , φ) expands in poloidal and toroidal vector fields, S i (r) and T i (r), with Y m i ℓ i (θ,φ) being spectral harmonics, where l and m describe the latitudinal and longitudinal mode number, respectively. The measured fields are used to constrain radial profiles of S i (r) and T i (r) that are approximated by low-order splines in radius. The reconstructions are truncated to a spectral resolution of l = 7. The geometry is found experimentally to be statistically axisymmetric; thus only m = 0 harmonics are used. From Equation (4) the current is found as

J =-∇×∇ 2 S r + ∇×∇×T r.
(5)

In addition B can be directly computed from the kinematic response of the system to an externally applied magnetic field, assuming that the velocity field is known and that the turbulent emf is a small correction. The predicted B from the kinematic calculation is also consistent with the Hall probe measurements. (thick black line in E θ ) as the smoothing parameter. All three signals show high fluctuation level compared to the mean values, i.e., σ r / V r ≈σ φ / B φ ≈30% for the radial velocity and longitudinal magnetic field components shown. In the case of the latitudinal component of the emf E θ , the fluctuations are almost 3.5 times higher compared to the mean value. Similar very strong turbulent fluctuations and small non-zero mean values are also found for the two remaining components E r , E φ (not shown here).

DIRECT MEASUREMENTS OF THE TURBULENT emf

A turbulent emf had previously been inferred from the meanfield response to applied seed fields in MDE. A combined analysis of magnetic data, taken in MDE, and laser Doppler velocimetry data, recorded in a similarly sized water experiment, yielded an estimate of E which was of the same order of magnitude as the mean flow induction V × B (Spence et al. 2007). There was strong evidence that velocity fluctuations generating E occurred at the system scale. In situation studied experimentally here, the system-scale fluctuations have been strongly reduced by introduction of an equatorial baffle (Kaplan et al. 2011). The equatorial baffle reduces the amplitude of the largest-scale turbulent eddies, but it also now provides the scale separation needed for the mean-field ansatz used in Equation (3) to describe the role of turbulent fluctuations. With the 3D emf probe all three components of V , B, and E are now directly measured. The ratio V × B /E 10 for 0 < Rm tip < 160 confirms that the turbulent emf is more than an order of magnitude smaller compared to the mean-field emf after installing the equatorial baffle.

Table 1

The Measured Values of ṽ, τ corr ,β,andα fit for Different Rm tip used in Figure 4(a) to Calculate the Corresponding Terms of Equation ( 3)

Rm tip vτ corr βα fit (m s -1 )( s )( m 2 s -1 )( m s -

ENHANCED TRANSPORT BY TURBULENCE

Figure 3(a) shows the angle between the unit vector of the turbulent emf with respect to the unit vector of the magnetic field (E, B ) (dashed line) as well as with respect to the unit vector of the corresponding local current (E, J ) (solid line), where J is obtained from Equation ( 5). With increasing Rm tip , E becomes almost orthogonal to B and anti-parallel to J . The 3D vector diagram in Figure 3(b) (shown for Rm tip = 140) indicates that the measured turbulent emf E (thick red arrow) almost entirely describes a vector anti-parallel to J (black arrow). The figure also includes the results of a numerical simulation, E sim (thin light red arrow), in which the turbulent emf is computed using a parallelized version of the magnetohydrodynamic DYNAMO code (Reuter et al. 2011), which solves numerically the induction equation and Navier-Stokes equation in an incompressible electrically conducting fluid including magnetic diffusivity. An accurate numerical copy of the two counter-rotating impellers, as used in MDE, generates a similar two-vortex flow collinear to the motor axis. The numerical simulations are done at Rm = 40 and dipolar seed fields of 20 G. Time traces of the magnetic and velocity fluctuations are given at the position of the 3D emf probe. The turbulent emf E sim ,

E sim ≡ v sim × b sim , (6)
as depicted in Figure 3(b), is obtained by using the same techniques as for the experimental data. Considering that the fluid Reynolds number for the simulations (Re 2000) is several orders of magnitude lower compared to MDE (Re ≈ 10 7 ), it is remarkable that the resulting simulated turbulent emf E sim aligns well with the measured one. Hence, measurement and numerical simulation clearly show that the turbulent emf, in the kinematic regime considered, is dominated by the second term on the right-hand side of Equation ( 3), which describes the β-effect.

This conclusion is also confirmed quantitatively. The directly measured velocity fluctuations v and autocorrelation times τ corr give β = v 2 τ corr /3 for different Rm tip (Table 1). Together with the current determined from the magnetic reconstruction, one can compute the diffusive term βμ 0 J in Equation (3), which is showninFigure4(a), red. One observes that, within the accuracy of the estimation, this diffusive term accounts for the majority of the directly measured turbulent emf ||E|| (Figure 4(a), black). A rough estimate for the local α-effect due to flow helicity (the first term in Equation ( 3)) can be determined by a linear fit of the longitudinal component E φ as a function of the locally measured longitudinal component B φ for different Rm tip (Figure 4(b) and Table 1). Including the estimated α-effect in Equation (3), represented by the blue squares in Figure 4(a), does not change the overall trend and confirms the dominance of the β-effect. It is noted that up to the present day no self-excited magnetic field 1).

(A color version of this figure is available in the online journal.) growth has been observed in MDE. In order to achieve dynamo action further experiments including stronger motors for driving the impellers and extra sets of rotatable baffles (in addition to the previously mentioned equatorial baffle) to optimize the ratio between poloidal and toroidal flows and to increase the flow helicity are planned.

SUMMARY AND OUTLOOK

In summary, this work presents direct observations of the 3D turbulent emf in a turbulent flow of liquid sodium. The measurements support the ansatz of an enhanced eddy resistivity predicted by simple mean-field theory models. This is the first time such a measurement has been possible in a laboratory setting, as described here. Future studies of the turbulent emf using multiple arrays of 3D emf probes are in preparation. These and other forthcoming experimental measurements of the global structure of MHD turbulence will provide detailed insight into key questions addressing the effect of large-scale and small-scale helicity (α-effect) and the role of molecular diffusivity in mediating the turbulent transport of magnetic fields.

The measurements discussed here were taken in a laboratory experiment, but the general nature of the findings has great impact for our understanding of the turbulent transport of vector quantities within astrophysical objects. In particular, for isotropic turbulence where the magnetic stresses are weak, these measurements strongly support simple mean-field closure methods that have been of vital importance in the modeling of magnetic transport in astrophysics, particularly for the dynamo problem (Blackman & Field 2002;Brandenburg & Subramanian 2005b). The turbulent transport of magnetic fields likely also plays a vital role in planet and star formation, and in MHD dynamics on galactic scales. Just such an enhanced magnetic diffusion may be essential for setting the solar activity cycle (Charbonneau 2010) and also the rate of star formation in the magnetic interstellar medium (Gaensler et al. 2011). Turbulence in astrophysical settings can be significantly non-isotropic or dominated by magnetic stresses, and those parameter regimes are a direction for future experiments.

Figure 1 .

 1 Figure 1. Madison Dynamo Experiment (MDE). (a) Picture of experimental device and schematic with important features and main diagnostics labeled. (b) Three example time traces from an experimental run for the radial flow (top), longitudinal magnetic field (center), and latitudinal turbulent emf (bottom). Mean values and standard deviations are indicated. (A color version of this figure is available in the online journal.)

Figure 2 .

 2 Figure 2. Picture and schematic of the newly designed 3D emf probe.(A color version of this figure is available in the online journal.)

P 1 P

 1 2 (V meas × B pm ) • dl yields a direct measurement of the fluctuations and time-averaged mean values of the flow components (von Weissenfluh 1985; Noskov et al. 2009) V meas = V + v for frequencies smaller than the system-based cutoff frequency f cs ≫ 1/(μ 0 σd 2 )( N o s k o v et al. 2009).

Figure 1

 1 Figure 1(b) shows three example time traces for the radial flow (top), longitudinal magnetic field (center), and latitudinal turbulent emf (bottom). The data were taken with an externally applied axial dipole field of 50 G at Rm tip = 160 (i.e., the maximal Rm given at the tip of the impeller). The corresponding mean values (time-averaged over 60 s) and standard deviations are given next to each time trace. The correlation of the fluctuating flow and magnetic field yields the turbulent emf directly. The component of the turbulent emf depicted is averaged by using the magnetic diffusion time τ σ = μ 0 σR 2 ≈ 3s

Figure 3 .

 3 Figure 3. Turbulent emf. (a) Angle of the unit vector of the turbulent emf E with respect to the current J (solid line) and the magnetic field B (dashed line). (b) Three-dimensional view of the unit vectors E (red), J (black), B (blue), and the result of a fully turbulent numerical simulation E sim (light red). (A color version of this figure is available in the online journal.)

Figure 4 .

 4 Figure 4. Mean-field electrodynamics interpretation. (a) Comparison of the β-effect-associated part of Equation (3) (red circles, red line) with the directly measured turbulent emf ||E|| (black crosses, black line). The blue circles represent Equation (3) by including an estimate for the local α-effect. (b) A linear fit of E φ (B φ ) yields an estimate of the local turbulent helicity α for four different Rm tip (Table1). (A color version of this figure is available in the online journal.)

Current address: Max-Planck-Institut für Plasmaphysik, Wendelsteinstr. 1, D-17491 Greifswald, Germany.

The authors thank Ingrid Reese, John Wallace, and Paul Brooks for technical assistance and probe work at the MDE. This project is funded by the National Science Foundation and is a member experiment of the Center for Magnetic Self Organization in Laboratory and Astrophysical Plasmas (CMSO). B.P.B. is supported in part by NSF Astronomy and Astrophysics postdoctoral fellowship AST 09-02004. CMSO is supported by NSF grant PHY 08-21899.


Franck P Vidal

Evelyne Lutton

Jean Louchet

Jean-Marie Rocchisani

Artificial evolution for PET and SPECT reconstruction

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

Introduction

Image reconstruction in tomography is an ill-posed inverse problem. This problem can be solved as an optimisation problem, and on such cases, evolutionary algorithms (EAs) have been proven efficient in general, and in particular in medical imaging. We focus here on tomographic reconstruction in PET.

Artificial Evolution

Evolutionary computing is a family of optimisation algorithms relying on Darwin's principles. In particular, it makes use of operators based on the biological mechanisms of natural evolution.

Standard PET reconstruction algorithms

Reconstruction methods in nuclear medicine are often divided into two classes: i) analytical methods, and ii)iterative statistical methods.

Analytical methods are based on continuous modeling and the reconstruction process consists in the inversion of measurement equations. The most frequently used algorithm is the filtered backprojection (FBP).

Statistical methods are based on iterative correction algorithms. These methods are relatively easy to model: 1. the reconstruction starts using an initial estimate of the image (generally a constant image), 2. projection data is computed from this image, 3. t h e e s t i m a t e d p r o j e c t i o n s a r e c o m p a r e d w i t h t h e measured projections, 4. corrections are made to correct the estimated image, and 5. the algorithm iterates until convergence of the estimated and measured projection sets.

There are different ways to implement these iterative methods. The main differences are about the computation of the projections, how the physics corrections (scattering, random, attenuation, etc.) are applied, and how the error corrections are applied in the estimated projections. Iterative methods include the most widely used techniques in SPECT and PET, such as the maximum-likelihood expectation-maximization method (ML-EM) and its derivative, the ordered subset expectation-maximization algorithm (OS-EM).

PET reconstruction using the fly algorithm

The algorithm that we present here follows the iterative algorithm paradigm. In preliminary studies, we introduced a cooperative coevolution strategy (or "Parisian evolution") called "fly algorithm" to minimize errors between the estimated projection data and the measured data. We showed that this approach can be used in SPECT reconstruction 1 and PET reconstruction 2,3,4 . Here, the searched distribution of radionuclides is modeled as a sample set of 3D points, the population of "flies". Each fly emits either photons or positrons depending on the image modality. The evolutionary algorithm evolves the position of flies using genetic operators to match reconstructed data with measured data.

The steps of the iterative method can be described as follows:

•Start with an initial guess. Initially, the flies' position is uniformly distributed within the volume defined by the scanner, or distributed depending on an initial estimate reconstructed using a fast simple back projection (SBP) performed on GPU using OpenGL. Each individual, or fly, corresponds to a 3D point.

•To produce estimated projection data, each fly mimics a radioactive emitter, i.e.a stochastic simulation of annihilation events is performed. For each annihilation event, a photon is emitted in a random direction. A second photon is then emitted in the opposite direction. If both photons are detected by the scanner, the corresponding LOR is recorded. The scanner properties (e.g.detector blocks and crystals positions) are m o d e l e d , a n d e a c h f l y i s producing an adjustable number of annihilation events. •The optimisation is performed using genetic operations. The fitness function used during the selection operation takes into account the discrepancies between the estimated projections and the measured projections (see [START_REF] Vidal | New genetic operators in the Fly algorithm: application to medical PET image reconstruction[END_REF] for details about the fitness metric). The regularization i s t a k e n c a r e b y t h e mutation operator. Multi-resolution is achieved thanks to our mitosis operator. When the reconstruction is optimum at the current resolution, an automatic mitosis is triggered to double the population size, hence improve the resolution (see [START_REF] Vidal | New genetic operators in the Fly algorithm: application to medical PET image reconstruction[END_REF][START_REF] Vidal | Threshold selection, mitosis and dual mutation in cooperative co-evolution: application to medical 3D tomography[END_REF] for details about our specific genetic operators).

•The algorithm iterates until the convergence of the estimated data with the measured data, i.e.the spacial concentration of flies will correspond to an estimate of the radionuclides' concentration.

Results

In [START_REF] Vidal | PET reconstruction using a cooperative coevolution strategy in LOR space[END_REF][START_REF] Vidal | Artificial evolution for 3D PET reconstruction[END_REF], we showed the ability of the early version of the algorithm (i.e. without taking advantage of some specific genetic operators we designed in [START_REF] Vidal | New genetic operators in the Fly algorithm: application to medical PET image reconstruction[END_REF][START_REF] Vidal | Threshold selection, mitosis and dual mutation in cooperative co-evolution: application to medical 3D tomography[END_REF]) to reconstruct simple 2D objects at low resolution. In [START_REF] Vidal | New genetic operators in the Fly algorithm: application to medical PET image reconstruction[END_REF], results at higher resolution are presented, as well as the fully-3D reconstruction of an object with a complex shape. This section presents new results, obtained using our specific genetic operators, with more sophisticated numerical phantoms of growing complexity.

Conclusion

New results of positron emission tomographic reconstruction using a specific cooperative co-evolution scheme based on the fly algorithm have been presented. It demonstrated the ability of the algorithm to reconstruct images using input data that corresponds to standard phantom models (the Standard Jaszczak phantom) and anatomically realistic models (cardiac and brain). However, the reconstruction of hot regions seems better than the reconstruction of cold areas; this needs to be addressed. Further work will include a concurrent study with the OS-EM algorithm and a quantitative analysis of the results. Further work will also include the correction of photon attenuation and Compton scattering in the modeled system matrix.

Flies for PET: an Artificial Evolution Strategy for Image Reconstruction in Nuclear Medicine

F

  . P. Vi d a l a,b , É. Lutton b , J. Louchet c , and J.-M. Rocchisani d a University of California, San Diego, CA b INRIA Saclay --Île-de-France, France c Artenia, Châtillon, France d Paris XIII University, UFR SMBH & Avicenne hospital, Bobigny, France

We present an iterative algorithm based on an evolutionary approach for image reconstruction in nuclear medicine. Our method is based on a cooperative coevolution strategy (also called Parisian evolution): the "fly algorithm". Each fly is a 3D point that mimics a positron emitter. The flies' position is progressively optimised using evolutionary computing to closely match the data measured by the imaging system. The population of flies approximates the radioactivity concentration. We have developed new genetic operators that have been proven to be more efficient than state-of-the-art operators used in evolutionary computing 5 . To speed-up computations, the reconstruction is automatically performed at progressive resolution.


Florent Lobligeois
email: florent.lobligeois@irstea.fr

Vazken Andréassian

Charles Perrin

Cécile Loumagne

Investigating the influence of rainfall spatial variability on flow simulation

Introduction

Weather radar is considered to be helpful for hydrological forecasting since it provides rainfall estimates with high temporal and spatial resolution. However, it has long been shown that quantitative errors inherent to the radar rainfall estimates greatly affect rainfall-runoff simulations. As a result, the benefit from improved spatial resolution of rainfall estimates is often limited for hydrological applications [START_REF] Biggs | A comparison of gauge and radar precipitation data for simulating an extreme hydrological event in the Severn Uplands, UK[END_REF][START_REF] Borga | Accuracy of radar rainfall estimates for streamflow simulation[END_REF][START_REF] Delrieu | Weather radar and hydrology[END_REF][START_REF] Emmanuel | Evaluation of the new French operational weather radar product for the field of urban hydrology[END_REF][START_REF] Krajewski | RADAR-Rainfall Uncertainties[END_REF]) compared to the use of traditional ground networks. Recently, Météo-France developed a rainfall reanalysis over France at the hourly time step over a 10-year period combining radar data and rain gauge measurements: weather radar data were corrected and adjusted with both hourly and daily rain gauge data [START_REF] Guéguen | Spécifications des traitements radar en vue d'une ré-analyse des lames d'eau[END_REF].

Here we propose a framework to evaluate the improvement in streamflow simulation gained by using this new high resolution product. The originality of our work is that we do not work on synthetic (simulated) streamflow, but with actual measurements, on a large set of 181 French catchments representing a variety of characteristics. The rainfall-runoff model is calibrated over a 5-year sub-period and evaluated on the complementary sub-period in validation mode. The results are analysed by catchment classes based on catchment area and the spatial variability of precipitation for various types of rainfall events.

Rainfall reanalysis

The rainfall reanalysis used in this study was recently developed by Météo-France and merges weather radar and rain gauges data (as illustrated in Figure 1). It provides hourly rainfall estimates at the 1-km² resolution, over a ten-year period (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) and at the hourly time step. This reanalysis is considered by Météo-France the best rainfall estimate currently available over the French territory.

+ + =

Figure 1: The rainfall reanalysis data set

Study area

A large set of 181 French catchments (see Figure 2) have been selected to run rainfall-runoff simulations. They represent a wide variety of hydrometeorological conditions, ranging from oceanic to Mediterranean. The characteristics of rainfall events on these catchments are also varied, with stratiform and convective events with a large range of intensities. Hourly discharge data at the basin outlets are available over the ten-year period of the rainfall reanalysis. The selected catchments are not influenced by snow and were chosen to be under the hydrologic visibility of weather radars.

Figure 2: The study area

Hydrological rainfall-runoff model

The rainfall-runoff model is the semi-distributed version of the GR4 lumped model [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]. GR4 is a parsimonious conceptual model which computes the discharge at the outlet of the catchment using precipitation and potential evapotranspiration data. Four model parameters need to be calibrated (Figure 3a). In the semi-distributed version, the catchment is divided into sub-basins. The outflow at the outlet of each sub-basin is calculated by a GR4 lumped simulation. Then it is routed to the catchment outlet by the lag routing method, which was found to provide a satisfactory level of efficiency in this study compared to more sophisticated methods. To limit model complexity, the parameters of the hydrological model were constrained to be the same on all sub-basins. Therefore, only five parameters have to be calibrated: the four parameters of the GR4 lumped model and the celerity parameter of the channel routing method (Figure 3b).

Methodology

The sensitivity of streamflow simulations to the spatial resolution of rainfall estimates was investigated by performing semidistributed rainfall-runoff simulations. Five levels of catchment splitting were studied: catchments were divided into 1, 2, 4, 8 and 16 sub-basins respectively (Figure 4). The rainfall input data were averaged over each sub-basin. Then, for each level of catchment splitting, the rainfall-runoff model parameters were calibrated against observed discharge data at the outlet of the catchment. Finally, the computed outflow was compared with the observed outflow at the catchment outlet in validation mode.

Results of the rainfall-runoff simulations

The model versions were tested using a split-sample test. Simulations were evaluated on the 10-year period in validation mode. For each of the 181 French catchments, thirty flood events were analyzed to investigate the impact of spatial precipitation inputs on flow simulations. As an example, two flood events for the Hérault catchment at Gignac (1430 km²) are presented in Figure 5. During the first flood event (Figure 5a), the spatial variability of precipitation is well taken into account when the level of catchment splitting increases and, as a result, the quality of flow simulation is improved. At the opposite, for the second flood event (Figure 5b), flow simulations are very similar regardless of spatial precipitation forcing. The spatial pattern of precipitation is better represented in the hydrological model when the number of sub-basins increases. However, it does not necessarily yield better flow simulations. We will try to understand and explain in which conditions the spatial rainfall variability does have an impact on flow simulation.

Figure 3 :

 3 Figure 3: The rainfall-runoff model

Figure 4 :

 4 Figure 4: Example of five levels of catchment splitting for the Hérault catchment at Gignac (1430 km²).

Figure 5 :

 5 Figure 5: Example of two flood events for the Hérault catchment at Gignac (1430 km²)

Acknowledgment

The authors would like to thank Météo-France and SCHAPI for providing the rainfall reanalysis and streamflow data respectively, and assisting with the data conversion and compression of files.