Mustapha Bekkhoucha

PRECISION CONCERNING THE COMPLEMENTS OF THE "RIEMANN HYPOTHESIS PROOF"

Keywords: Mathematics Subject Classification. 11M26 Riemann zeta function, nontrivial zeros, Riemann hypothesis

It is about the infinite partition of th real interval (-∞, 1/2) associated with each complex ζ(s)-zero, or more precisely, with the two ζ * ′ (s)-zeros going with the latter.

One shows that this partition is regular, with the points of division k.

1 2 , k ∈ Z and k ≤ 1.

In addition, by looking further this question, it appears that certain parabolic forms could play a role in quantum mechanics, at the level of elementary particles and beyond.

Introduction

One establishes here the connection with a result given at the end of the complements. It is about the property that each zero ρ of ζ(s) is accompanied on the axis ℜ(s) = 1/2 by some points where ζ * (s) takes the value ζ * (ρ)(the last of these points in number ≥ 2) satisfying moreover ζ * ′ (s) = 0.

In order to go further, and in particular to specify the partition of the interval (-∞, 1/2) in partial intervals where ω must depends on whether it is one or the other of the ζ * ′ (s)zeros which is used, one start by making the simplest assumption, namely that the number of these zeros is 2, that their order of multiplicity is 1. Now, it is clear that the functioning of this set of points associated to ρ (let us call "motif") depending on the position of ω, as it was described at the end of the complements, is incomplete or insufficient. Indeed, in order to avoid that the angle Φ remains > -π, once reached the lower limit ω 1 of the first partial interval, because it would result from it an indetermination on ω, one has to assume that Φ becomes < -π, by putting Φ = -π +β with β < 0. This implies that the situation relating of the respective tangents to the ζ * -curves images of the half-straights ωS ′ , ωS 1 , changes direction completely, so that, in ω 1 , the roles of the upper and lower half-plane were exchanged.

That is the indication of intervention of a second "motif", which would have function, in synchronizing with the first "motif", and which, wile the first sends a wave in the upper half-plane, sends a wave in the lower half-plane, waves which will cross in ω 1 .

In a more precise way, one will consider a new entity (let us call it "bimotif") by associating the two motifs : the first with its two remarkable points

s 11 = 1 2 + iτ 1 , s 12 = 1 2 + iτ 2 , τ 1 > τ 2 > 0
the second, with the two corresponding points

s 21 = γ 2 -iτ 1 , s 22 = γ 2 -iτ 2 , γ > 1.

The intrication of the two motifs in the bimotif

It is supposed from now that the orthogonal projection ω 0 of the double zero z 0 of ζ * (s) on the real axis, is not other that 1/2. Then one assign the origin 0 as lower limit ω 1 of the first interval partial of the partition of (-∞, 1/2) in the initial functioning of the motif. Here why :

Let us call ω the point describing the segment of origin 1/2, and, end 0, ω ′ the point describing the segment of origin γ/2, and end 0, by dividing it in the ratio that ω divides the segment [1/2, 0]. By putting

1 2 -ω = τ 1 tan α , γ 2 -ω ′ = τ 2 tan α ′ that result in τ 1 tan α = 1 γ τ 2 tan α ′ .
One knows (cf [START_REF] Bekkhoucha | Riemann hypothesis proof[END_REF] lemma1.p8) that the upper vertical half-straight of abscissa ω 0 (= 1/2) is the first, while coming by the right, to have its ζ * -image which has points in the lower half-plane, and that the point crossing of the real axis is the origin 0. The transposition of show that the lower half-vertical of abscissa γ/2 is the first, while coming by the right, to have its θ-image which has points in the upper half-plane, and which the point of crossing the real axis is the origin.

Thus, the two waves: ζ * (s) emitted by the first motif in the upper half-plane, c.ζ * (sγ) emitted by the second motif in the lower half-plane, cross at the point O, c being already given according to γ, such that

ζ * ( 1 2 ) = c.ζ * ( 1 2 -γ)
(the two waves coincide in 1/2). One also has, according to what precedes

ζ * (0) = c.ζ * (0 -γ)
(by making expressly the point z 0 of the first motif to intervene, and also its homologous z ′ 0 in the second motif).

These two relations are one of the elements of the intrication of the two motifs constituting the bimotif. Another element of this intrication is that the values α 0 , α ′ 0 of α, α ′ , for ω = 0, are complementary. It is easy to see that is equivalent to OS 11 ⊥OS 22 , or S 11 OS 22 = a right angle, or 2 S 11 OS 22 =a flat angle. (As this last relation suggest it, the condition posed is a manner of establishing a connection between the bimotif when ω comes on 0, and the motif isolated initially, when Φ reaches its limit value Φ 1 = -π, what, by taking again the initial notations, corresponds to 2 s 1 ω 1 s 2 =flat angle (cf [START_REF] Bekkhoucha | Riemann hypothesis proof[END_REF]

last line of p 21))

The intrication is confirmed by putting

4τ 1 τ 2 = γ 1 4 + τ 2 1 = γ 2 4 + τ 2 2 or τ 1 = γ 2 , τ 2 = 1 2

The antibimotif

Returning once again to the functioning of the motif isolated, one saw how the first phase starts at 1 2 and finishes in 0, by integrating this functioning in that of the bimotif. The two motifs act in the same direction; from the right hand towards the left.

To be able to pass on the left from 0, as planned for the second phase, it will be necessary to associate to the bimotif, the bimotif which acts on contrary direction (like the manner of the alternating current). It will be called antibimotif.

Functioning of the system:

It is necessary to have γ = 1 to make functioning the system. It is indeed under that condition, that the two characteristic triangles (rectangles-isosceles), have in common their

two apex 1 2 (1 + i), 1 2 (1 -i) (thus forming a square of center 1 2
, and whose two other apex are 0 and 1). So, one can consider that wave emitted by bimotif (functioning between 0 and 1 2 ), and the wave emitted by the antibimotif (between 1 2 and 1), are the same waves functioning between 0 and 1.

Because the system, at the end of the cycle, returns to its initial state, it is possible to repeat this cycle indefinitely throughout the real axis. Let us express the system bimotifantibimotif (with its two right-isosceles triangles) corresponding to γ in a same context : the one of complex plane network.

Because the symmetrical role of the upper and lower half-planes in the bimotif and in the antibimotif, because also the shape of the intrication of the two motifs in these latter, one is leading to use the orthogonal network generated by 1 + i 2 and 1 -i 2 . Now, in this representation, the intrication in the system, corresponding to γ = 1, corresponds to the central symmetry relatively to the origin. Then, one adopts this same correspondence to define the intrication of the two components of the system for the general case with γ = an appropriate integer. Moreover, the fact of generating the same network shows that they emit the same wave. γ must be such that

2 1 + γ 2 /2 × √ 2 2 = q (integer) or 1 + γ 2 = 2q 2 q ∈ N Clearly, The points k. 1 2
, k ∈ Z are center of symmetry for the network, and therefore the points of division for the real axis, what answers the question asked initially about the partition of the interval (-∞, 1/2).

Note about the introduction of the lattice generated by the "bimotif"

Whereas initially one considered the introduction of the case γ = 1 only like borderline case, it seems that this case is the normal case at certain point of view. Indeed, one sees well, according to the calculations of page 3, that the general case γ > 1, yields a distortion of the geometry of the space (represented by one of its directions, the axis ℜ(z) = 1/2) with τ 2 = 1/2 too lower relatively to τ 1 = γ/2. The rotation carried out to bring the axis of symmetry of the characteristic triangle to the horizontal position like for the case γ = 1, would be thus a manner of correcting the preceding distortion, and, in the same time, of preserving the symmetry of the role played by the upper and lower half-plane. One will thus start with : 4.1. The study of the case γ = 1. The two waves discussed before, are then ζ * (s) and c.ζ * (s -1) (with the value of c corresponding to γ = 1). It was shown that these two waves emitted respectively in the upper half-plane and the lower half-plane, cross the real axis while meeting in the origin. It is specified now that they cross with vertically directed and opposite speeds. Indeed, the half-line Oz 0 , having polar angle π/4, one saw in section 1 (cf [START_REF] Bekkhoucha | Riemann hypothesis proof[END_REF] p12) that the "retrogression" tangent in ζ * (z 0 )(= 0) by the ζ * -image of the half-line, has a polar angle =

π 4 × 2 = π 2 .
The same goes symmetrically relatively to the real axis, except that π/2 must be replaced by -π/2. By symmetry relatively to 1/2, there is a similar result for the fundamental antibimotif. The corresponding waves cross in 1, with opposite speeds and directed vertically. Thus, for the states of the waves in 0, 1/2, 1, it seems the one which cut across orthogonally the real axis in these points with this order (or the conjugated wave for the opposite order), by passing through 1/2, with the angle ± π 4 . Moreover, the state of this wave is the same in 0 and 1, which suggest, using the translation of Z, to extend it to all points k. 1 2 , k ∈ Z. In fact, this extension can be obtained by the rational function ∆(z) = (-z 2 (1 -z 2 )) -2 (with exponent -2 like for the parabolic form of weight 12: in fact the parabolic form ∆ is given, with q = exp 2iπz, by

∆(q) = (2π) 12 q ∞ n=1 (1 -q n ) 24
but the two expressions differ only by a multiplicative constant, since they concern modular forms of the same weight 12).

Let us finish the study of the fundamental case, while returning to the property sought initially, in connection with the partition of the interval (-∞, 1/2). It is clear that the symmetries mentioned above, or only the subgroup of translations (k ∈ Z) are sufficient for leading that the fundamental square of apex 0, 1,

-(1 + i) 2 , ( 1
-i) 2
yields by its images, copies throughout the real axis, and in particular, one thus obtains the partition of (-∞, +∞) by the points k. 1 2 , k ∈ Z. Also, let us note that one provide not only the symmetry of the role played in the system fundamental bimotif-antibimotif, by the two half-planes, but also the symmetry between the two directions of the real axis.

Study of the general case.

It is sufficient to suppose that the numbers γ and q = √ 2 2 1 + γ 2 are integers, in order to make the lattice generated by the characteristic triangle of the bimotif (once its axis brought back to horizontal) a sub-lattice of the fundamental lattice.

One saw before that this condition is equivalent to (q, γ) is an integer solution of the "diophantienne" equation 2q 2 -γ 2 = 1, or q √ 2 -γ is invertible in the euclidean ring Z √ 2, and consequently is of the form (-1 + √ 2) n , n ∈ N. It results from this that the sought solutions are an increasing sequence γ n q n of rational numbers tending to √ 2.

A special intrication between the fundamental level (corresponding to ∆)

and each lattice corresponding to the modular forms

G 2 , G 3 , G 4 , G 5 , G 7
All start with the wave ∆, operating in a space-time of dimension 2, with the space of dimension 1. The wave ∆G 2 operates in a space-time of dimension 3, with the space of dimension 2, but G 2 operates in a space-time of dimension 2 (thus with the space of dimension 1) and can consequently be identified with ∆ under some condition, more exactly, when the corresponding lattice is a sub-lattice of the lattice corresponding to ∆. This intrication is repeated twice, like the number of spatial dimensions. Generally speaking, the wave ∆G k , k = 2, 3, 4, , 5, 7 operates in a space-time of dimension > 2, whereas G k operates in a space-time of dimension 2, and the intrication appears when the lattice corresponding to this last is a sub-lattice of the fundamental lattice. This intrication is repeated as many as the number of spatial dimensions.

In terms of the physics of elementary particles, and even in terms of entities of higher scales, let us agree to associate ∆ on the scale of the motif (space of dimension 1), ∆G 2 on the scale of the proton-neutron (space of dimension 2 : R 2 ), ∆G 3 on the scale of the nucleus (space of dimension 3 : R 3 ), ∆G 4 on the scale of the atom (space of dimension 4: R 4 = R 3 × R, R 3 for the nucleus, R for the orbits of the electrons), ∆G 5 on the scale of the star (space of the atoms with a new dimension for the gravity), ∆G 7 on the scale of the galaxy (space of dimension 7 : 3 for the galaxy, and 3 probably for the supermassif black hole which is associated to it, and 1 for the gravity).

Consequences of the special intrications

(1) Intrication appearing thanks to the property that the lattice associated to G k is a sub-lattice of that associated to ∆, since the symmetries of a sub-lattice are those of the lattice, the answer to the question asked initially will be the same: the time is marked by the sequence of points of the real axis k. 1 2 , k ∈ Z . Thus, for all the scales, it is the same time. (2) On the scale of the star, or the galaxy, since a cell representing the sub-lattice is made by cells representing the lattice of ∆, the electrical current is annihilated inside, because each of the interior side of the elementary cells is covered in one direction and the opposite direction. There remains only the current of one of edge. The interior matter having lost its electric and luminous properties, and became this black matter of which is question in the cosmos. (3) On the scale of the galaxy, while returning to the antibimotif-bimotif complex which is associated to it (after switching the axis of the bimotif on the real axis, and the replacement of symmetry relatively to 1/2, by symmetry relatively to 0), one saw higher that the position in space, and the time q n made to reach this distance are connected by the relation 2q

n < γ n < q n √ 2) and that q n √ 2 -γ n = ( √ 2 -1) n . It results from this last relation that √ 2 - γ n q n = ( √ 2 -1) n q n → 0 while decreasing.
Thus γ n q n is an increasing sequence, and consequently whereas q n is measured with a uniformly passing time, it is not similar for γ n : the measuring unit of the space does nothing but increase, thus giving the impression of an accelerated expansion of the space, unless one suppose that the space is marked previously with a speed exceeding the one of the light. (4) Solar neutrinos: the deficit of solar neutrinos in the detectors could be due to the special intrication of the fundamental lattice (corresponding to ∆) and the sub-lattice (G 5 ). Indeed, the last which appears at long distances, sees its cells (assemblies of elementary cells) to lose the electromagnetic properties of interior cells, as explained precedently (black matter), as that only the fraction of elementary cells not belonging to these last, can react with the detector.

Conclusion

The development which one gave to the initial question, shows that the macroscopic world is already annouced in the microscopic world. Like this, the black matter, which is revealed only at the level of the star and the galaxy, is already present at the level of the motif, through the existence of a sector (of apex ω), between two successive motifs, sector which, contrary to the preceeding sector, is not corresponding to any property of the matter, except for this form of the gravity that goes with the appearance of each sector and point out by the term 2kπi in (ln ω,k ((1 -s)ζ * (s) + 2kπi)) -1 in order to bring the system down the fundamental state (cf. [START_REF] Bekkhoucha | Riemann hypothesis proof[END_REF] p16).

Figure 1 this

 1 Figure 1 this result to the function in (x ′ , y ′ ): θ(x ′ , y ′ ) reciprocal image of ζ * (x, y)(= ζ * (s)) by the dilatation of the real half-axis, of ratio γ x ′ = x/γ , y ′ = y

  rising from the points symmetrical of s 11 , s 22 , s 12 , s 21 .

Definition 1. The antibimotif associated to the bimotif, with its motifs of respective abscissas 1 2 , γ 2 , carrying its couple of remarkable points (s 11 , s 22 ), (s 12 , s 21 ), is the bimotif symmetrical relatively to the point 1 2 . The waves which are associated to it are thus ζ * (1-s), c.ζ * (1-s-γ),

  2 n -γ n = 1 (what implies the double inequality q


Student Member, IEEE Andrei Anghel

Member, IEEE Gabriel Vasile

Member, IEEE, Cornel Remus Cacoveanu

Cornel Ioana

Member, IEEE Silviu Ciochina

Short-range wideband FMCW radar for millimetric displacement measurements

Keywords: Frequency Modulated Continuous Wave (FMCW), Interferometry, Synthetic Aperture Radar (SAR), Nonlinearity Correction, High-Order Ambiguity Function (HAF), Time Resampling

come

I. INTRODUCTION

T HE frequency modulated continuous wave (FMCW) radar is an alternative to the pulse radar when the target range can be relatively small (below 100 m). In order to measure such distances with a pulse radar the switching time between transmission and reception should be at most tens of nanoseconds. For a FMCW radar the range information is provided by beat frequencies and each frequency corresponds to a target placed at a certain distance. Typical FMCW radar implementations have a homodyne architecture transceiver [START_REF] Meta | First demonstration of an X-band airborne FMCW SAR[END_REF], [START_REF] Charvat | Low-cost, high resolution X-band laboratory radar system for synthetic aperture radar applications[END_REF] which limits the performances for shortrange applications. In the case of targets positioned near the radar, the beat frequency is small and can be situated in A. Anghel is with the Telecommunications Department, University Politehnica of Bucharest (UPB), 060032 Bucharest, Romania (e-mail: andrei.anghel@munde.pub.ro), and also with the Grenoble Image sPeech Signal Automatics Laboratory (GIPSA-Lab), CNRS / Grenoble INP, 38402 Grenoble, France.

G. Vasile and C. Ioana are with the GIPSA-Lab, CNRS / Grenoble INP, 38402 Grenoble, France (e-mail: gabriel.vasile@gipsa-lab.grenoble-inp.fr; cornel.ioana@gipsa-lab.grenoble-ino.fr).

R. Cacoveanu and S. Ciochina are with the Telecommunications Department, UPB, 060032 Bucharest, Romania (e-mail: cacoveanu@munde.pub.ro; silviu@comm.pub.ro).

the frequency band affected by the classical problems of the homodyne architecture (DC offset, self-mixing and 1/f noise) [START_REF] Besser | Practical RF Circuit Design for Modern Wireless Systems[END_REF]. Another problem of the FMCW transceiver is that the voltage controlled oscillator (VCO) adds a certain degree of nonlinearity which leads to a deteriorated resolution by spreading the target energy through different frequencies [START_REF] Meta | Range nonlinearities correction in FMCW SAR[END_REF]. This issue is usually solved either by hardware [START_REF] Pohl | Ultra high resolution SAR imaging using an 80 ghz FMCW-Radar with 25 ghz bandwidth[END_REF], [START_REF] Lowbridge | A low cost mm-wave cruise control system for automotive applications[END_REF] or software [START_REF] Vossiek | Novel FMCW radar system concept with adaptive compensation of phase errors[END_REF], [START_REF] Meta | Signal processing for FMCW SAR[END_REF] approaches.

This paper presents a short-range FMCW X-band radar based on a heterodyne architecture of the transceiver which eliminates the low frequency self-mixing spectrum [START_REF] Anghel | Shortrange FMCW X-band Radar Platform For Millimetric Displacements Measurement[END_REF] and reduces the noise bandwidth. In other works like [START_REF] Charvat | A through-dielectric ultrawideband (uwb) switchedantenna-array radar imaging system[END_REF], [START_REF] Charvat | A through-dielectric radar imaging system[END_REF], [START_REF] Charvat | A low-power high-sensitivity x-band rail sar imaging system [measurement's corner[END_REF], [START_REF] Charvat | A Low-Power Radar Imaging System[END_REF] the heterodyne architecture is used for S-band and X-band FMCW transceivers in order to build a hardware range-gate based on narrow-band communication filters. The range-gate is used either to eliminate from the beat signal some powerful reflections that can reduce the receiver's dynamic range [START_REF] Charvat | A through-dielectric ultrawideband (uwb) switchedantenna-array radar imaging system[END_REF], [START_REF] Charvat | A through-dielectric radar imaging system[END_REF] or just to increase the sensitivity in the range swath of interest [START_REF] Charvat | A low-power high-sensitivity x-band rail sar imaging system [measurement's corner[END_REF].

The VCO nonlinearity problem is solved using a correction algorithm designed for wideband nonlinearities that can be described by a polynomial expression. The presence of polynomial nonlinearities leads to a multi-component polynomialphase beat signal. The coefficients of the polynomial-phase signal (PPS) are estimated using the high-order ambiguity function (HAF) [START_REF] Wang | On the use of high order ambiguity functions for multicomponent polynomial phase signals[END_REF] on a reference response which can be either a delay line or a high reflectivity target whose propagation delay should be roughly known. Afterwards, with the estimated coefficients, the nonlinearity correction function is built and applied through a time resampling procedure [START_REF] Anghel | FMCW Transceiver Wideband Sweep Nonlinearity Software Correction[END_REF].

The nonlinearity correction algorithm we propose in this paper differs from previous works in two ways. On one hand, a typical nonlinearity estimation method (used for example in Vossiek's work [START_REF] Vossiek | Novel FMCW radar system concept with adaptive compensation of phase errors[END_REF]) based on the determination of the instantaneous phase of a precision radar reference path is valid only for a single component response, while the HAFbased estimation can extract the nonlinearity coefficients from a multi-component response if there is one highly reflective target relative to other scatterers. On the other hand, the correction method proposed in [START_REF] Meta | Range nonlinearities correction in FMCW SAR[END_REF] by Meta et al. requires a multiplication between the beat frequency signal and the nonlinearity term from the transmitted waveform. This implies an oversampling of the beat signal in order to satisfy the Nyquist condition for the nonlinearity term bandwidth and consequently this method is not well suited for large band-width nonlinearities (up to gigahertz). In our work, both the estimation and the correction of the nonlinearity are made using only the beat frequency signal, so the bandwidth of the nonlinear term from the transmitted signal doesn't impose the sampling rate.

The X-band FMCW radar is tested with a number of range profiles and SAR images of a few short-range metallic targets. The displacements are computed from the interferometric phase and their absolute errors are below 0.1 mm in all tests.

This paper is organized as follows. In Section II the shortrange FMCW radar architecture is described. Section III presents the wideband nonlinearity correction algorithm. The polynomial-phase signal model is introduced first. Then, the HAF-based estimation method of the FMCW signal coefficients is discussed in Section III-A and the time resampling correction procedure is exposed in Section III-B. Simulation results prove the efficiency of the proposed correction algorithm in Section III-C. Section IV presents the measurement results: Section IV-A shows the validation of the nonlinearity correction algorithm and in Section IV-B the displacement measurements are presented. Finally, the conclusions are stated in Section V.

II. SHORT-RANGE FMCW RADAR ARCHITECTURE

The block diagram of the X-band short-range FMCW radar is shown in Fig. 1. The sweep signal is a linear tuning voltage with 100 ms period obtained from the signal generator of an USB oscilloscope. The discrete nature of the command signal leads to a stepped frequency modulation with an unambiguous beat frequency inversely proportional to the step duration [START_REF] Yang | Randomfrequency sar imaging based on compressed sensing[END_REF], [START_REF] Meta | Signal processing of FMCW synthetic aperture radar data[END_REF]. The resolution of the signal generator is 12 bits which for the maximum bandwidth of 4 GHz leads to an unambiguous range of over 100 m (where the behavior is FMCW type). The RF VCO block is a low-cost X-band VCO with 15% linearity (defined as the ratio between the maximum frequency deviation of the characteristic from linear and the total sweep bandwidth [START_REF]General specification for crystal controlled oscillator[END_REF], [START_REF] Brennan | Determination of sweep linearity requirements in FMCW radar systems based on simple voltagecontrolled oscillator sources[END_REF]) which provides the local oscillator (LO) signal. The high nonlinearity of this VCO can be software corrected and is no need of an expensive YIG-based VCO. For a linear tuning voltage, the RF VCO signal in a sweep period T p can be written as:

s LO (t) = cos 2π f 0 t + 1 2 α 0 t 2 + Φ nl (t) , (1)
where Φ nl (t) is the nonlinearity phase term. The intermediary frequency (IF) block is a direct digital synthesizer with adjustable frequency (50-250kHz). The transmitted signal consists of two different signals obtained by mixing the LO signal with the IF signal:

s T (t) = 1 2 cos 2π (f 0 + f IF )t + 1 2 α 0 t 2 + Φ nl (t) + 1 2 cos 2π (f 0 -f IF )t + 1 2 α 0 t 2 + Φ nl (t) . (2)
A part of the transmitted signal gets directly to the mixer from the receiver section through the couplers (C1 and C2) and the delay line. This reference path is used as power level reference and for calibrating the radar with the nonlinearity correction algorithm. In the receiver section, the reflected signal that comes from N different targets is a sum of delayed and attenuated versions of the transmitted signal s T (t):

s R (t) = N i=1 A i s T (t -τ i ), (3)
where τ i and A i are the propagation delay and amplitude corresponding to target i. The received signal is mixed with the LO and the resulting low frequency signal gets centered around the IF:

s IF (t) = 1 4 N i=1 A i cos 2π (f IF + α 0 τ i )t+ (f 0 -f IF )τ i - 1 2 α 0 τ 2 i + Φ nl (t) -Φ nl (t -τ i ) + cos 2π (f IF -α 0 τ i )t -(f 0 + f IF )τ i + 1 2 α 0 τ 2 i - Φ nl (t) -Φ nl (t -τ i ) . (4)
For short-range applications the delay is very small compared to the sweep period. In consequence the residual video phase (RVP) term [START_REF] Carrara | Spotlight Synthetic Aperture Radar: Signal Processing Algorithms[END_REF] can be neglected and the nonlinearity phase term difference can be approximated with the derivative multiplied with the delay. Under these assumptions, the IF signal can be rewritten as: This signal is filtered, amplified and afterwards sampled with 1 MHz sampling rate. The IF signal spectrum consists of two groups of spectral components placed symmetrically around the intermediary frequency as presented in Fig. 2. The analog band-pass filter (BPF: 25-500 kHz bandwidth) removes the low-frequency components (resulted from self-mixing and local oscillator leakage in the transmitted signal), improves the signal to noise ratio by reducing the noise bandwidth and acts as anti-aliasing filter. Additional digital filters can be applied to select an imposed range interval before mixing to baseband. The specifications of the wideband FMCW radar system are summarized in Table I.

s IF (t) ≈ 1 4 N i=1 A i cos 2π (f IF + α 0 τ i )t+ (f 0 -f IF )τ i + τ i Φ ′ nl (t) + cos 2π (f IF -α 0 τ i )t- (f 0 + f IF )τ i -τ i Φ ′ nl (t) . (5)
f IF Frequency DC offset f IF -f b,max f IF + f b,max f b,max 1 /f n o is e
Reference path Analog BPF Digital BPF Fig. 2. Intermediary frequency signal spectrum. The analog filter removes the low-frequency components, improves the signal to noise ratio and acts as anti-aliasing filter.

In order to shift the spectrum in baseband, the signal in ( 5) should be digitally multiplied with the sampled IF sinusoidal signal. The resulting beat frequency signal is (neglecting the 0.5 factors resulted from cosine multiplication):

s b (t) = N i=1 A i cos 2π f 0 + α 0 t + Φ ′ nl (t) τ i . (6)
If the range profile is computed as the Fourier transform of this beat signal, the nonlinearity terms spread the energy of each target and the resolution gets deteriorated. In order to avoid this shortcoming, the beat signal is processed with the nonlinearity correction algorithm described in the following section. Notice that the same beat signal as the one in ( 6) would be obtained for a homodyne architecture if we neglect the undesired low-frequency components, so the proposed nonlinearity correction algorithm can be applied to both heterodyne and homodyne architectures.

Note that in contrast to other implementations (like [START_REF] Charvat | A Low-Power Radar Imaging System[END_REF]) the proposed architecture has certain differences which start with the main reason for using the intermediary frequency (to avoid the homodyne problems, not to build a range-gate). Using a relatively large sweep period leads to a smaller frequency interval of the expected beat frequencies which implies a reduced noise bandwidth after the weighting window applied in computing the range profile (the equivalent noise bandwidth is around 15 Hz for 100 ms sweep period). Another observation is that both groups of spectral components around the IF are mixed to baseband and cumulated (not only one lateral sideband) which avoids using an image reject filter.

III. NONLINEARITY CORRECTION ALGORITHM

The slope of the frequency-voltage characteristic for some radio frequency VCOs may be reasonably approximated by a quadratic curve [START_REF] Brennan | Determination of sweep linearity requirements in FMCW radar systems based on simple voltagecontrolled oscillator sources[END_REF]. However, a more general approach is to assume a polynomial frequency-voltage dependence. This means that the nonlinearity phase term can be expressed as:

Φ nl (t) = 2π K k=2 β k k + 1 t k+1 . (7)
With this assumption, by introducing ( 7) in ( 6) the beat signal becomes a PPS:

s b (t) = N i=1 A i exp j2π f 0 + α 0 t + K k=2 β k t k τ i . ( 8
)
The correction algorithm proposed in this paper aims to turn this multi-component PPS into a sum of N complex sinusoids. In this way each target appears as a sinc function in the range profile. The algorithm consists of two steps: the estimation of the FMCW signal coefficients (linear chirp rate α 0 and nonlinearity coefficients β k ) using the high-order ambiguity function and the correction of the beat signal by time resampling.

A. Estimation of the FMCW signal coefficients

The estimation is based on the presence of a reference target response (with amplitude A ref and propagation delay τ ref ) in the FMCW signal. This particular PPS component can be extracted by bandpass filtering the beat signal around the beat frequency corresponding to τ ref . The filtered signal can be written as:

s b,f (t) = s b,f (t, τ ref ) + M m=1 s b,f (t, τ m ). ( 9
)
where M is the number of significant PPS components located near the reference response in the filter's pass band which cannot be eliminated. In the estimation, it is considered that the reference target is highly reflective relative to the remaining M components. Although the filtered signal has other components besides the reference response, the FMCW signal coefficients can be estimated using the high-order ambiguity function due to its ability to deal with multiple component PPS's if the highest order phase coefficients of the components are not the same [START_REF] Barbarossa | Product highorder ambiguity function for multicomponent polynomial-phase signal modeling[END_REF], [START_REF] Porat | Accuracy analysis of estimation algorithms for parameters of multiple polynomial-phase signals[END_REF]. This happens for the FMCW signal because each component is linked to a target with a certain propagation delay.

The starting point is the high-order instantaneous momentum (HIM), which can be defined for a signal s(t) as [START_REF] Wang | On the use of high order ambiguity functions for multicomponent polynomial phase signals[END_REF]:

HIM k [s(t); τ ] = k-1 i=0 s ( * i) (t -iτ ) ( k-1 i ) , ( 10
)
where k is the HIM order, τ is the lag and s ( * i) is an operator defined as:

s ( * i) (t) = s(t) if i is even, s * (t) if i is odd, ( 11
)
where i is the number of conjugate operator "*" applications. The high-order ambiguity function (HAF) is defined as the Fourier transform of the HIM.

If we assume a PPS model for the analyzed signal, i.e.:

s P P S (t) = A exp j2π k m=0 a m t m , (12)
the essential property of the HIM is that, the k-th order HIM is reduced to a harmonic with amplitude A 2 k-1 , frequency fk and phase Φ k :

HIM k [s P P S (t); τ ] = A 2 k-1 exp j 2π fk t + Φ k , (13)
where

fk = k!τ k-1 a k . (14)
So the HAF of this HIM should have a spectral peak at the frequency fk . Based on this result, an algorithm that estimates sequentially the coefficients a k was proposed in [START_REF] Peleg | Estimation and classification of signals with polynomial phase[END_REF]. Starting with the highest order coefficient, at each step, the spectral peak is determined, and an estimation value âk of a k is computed from [START_REF] Wang | On the use of high order ambiguity functions for multicomponent polynomial phase signals[END_REF]. With this value, the phase term of the higher order is removed:

s P P S (k-1) (t) = s P P S (k) (t) exp -j2π âk t k (15)
and the procedure repeats iteratively. A classical problem of this nonlinear method is the propagation of the approximation error from one higher order to the lower ones, but in the case of typical frequency-voltage VCO characteristics this effect is not critical because an approximation order of only 3 or 4 is required. Still, if a higher order is necessary a warpedbased polynomial order reduction as described in [START_REF] Ioana | Time-frequency analysis using warpedbased high-order phase modeling[END_REF] could be employed.

After applying this iterative algorithm to the FMCW reference signal and obtaining the polynomial phase coefficients, the linear chirp rate and the nonlinearity coefficients can be computed as:

α 0 = â1 τ ref β k = âk τ ref , k = 2, K. (16)

B. Time Resampling

Due to the fact that the frequency-voltage characteristic of a VCO is monotonous, for a linear voltage sweep the resulting polynomial phases of the beat signal components are monotonous functions for t ∈ [0, T p ]. Therefore, the beat signal in (8) can be rewritten as:

s b (t) = N i=1 A i exp {j2π [f 0 + α 0 θ(t)] τ i } , t ∈ [0, T p ], (17)
where

θ(t) = t 1 + K k=2 β k α 0 t k-1 (18)
is a monotonous function of time t, which can be interpreted as a new time axis. Hence, if the time axis is changed to θ, the beat signal becomes a sum of N complex sinusoids, which was the purpose of the correction algorithm. Moreover, in the context of radar detection, the highly correlated clutter from a nonlinear range profile gets decorrelated in a range profile computed for the new time axis. Notice that in the definition of θ the nonlinearity coefficients β k are normalized to the linear chirp rate α 0 which means that the exact value of the reference propagation delay is not needed. However, a rough value is required for the estimation part in order to extract the reference response.

From the implementation point of view, the beat signal is a digital signal s b [n] uniformly sampled at the moments t n , n = 0, N s -1, where N s is the number of samples. However, the samples s b [n] of the beat signal related to the moments θ n = θ(t n ) of the θ time axis lead to a non-uniformly sampled signal. It can be shown that the average sampling interval of θ is:

θ S = α α 0 T s , ( 19
)
where α is the mean chirp rate and T s the uniform sampling interval. According to [START_REF] Marvasti | Nonuniform Sampling:Theory and Practice[END_REF], for a nonuniformly sampled signal, the average sampling rate must respect the Nyquist condition.

For α > α 0 , this condition can be fulfilled if the beat signal is oversampled (the chirp rate in the origin and the average chirp rate typically have the same order of magnitude, so an oversampling of at most 10 is enough). If the signal is aliasfree, it can be resampled with an interpolation procedure (e.g. with spline functions) in order to obtain a uniformly sampled signal in relation with the θ time axis. Afterwards, the range profile is computed by applying the discrete Fourier transform to the resampled signal. The nonlinearity correction algorithm is summarized in the block diagram from Fig. 3.

s b (t) Reference response selection s b,f (t) HAF-based estimation of FMCW coefficients (α 0 , β 2 ,…, β K )              K k k k t t t

C. Nonlinearity Correction Algorithm Simulation

The range profiles of a FMCW radar based on an X-band VCO with 15% linearity were simulated. The chirp bandwidth was 4 GHz, the sweep rate 50 Hz and the sampling frequency 1 MHz. The responses of six stationary targets with different amplitudes were considered. The reference target was the one located at 50 m. The nonlinear and corrected range profiles are shown in Fig. 4. In the nonlinear range profile the targets can't be distinguished due to the overlapping of the frequency spread responses of each target. The correction algorithm enhances the -3 dB resolution up to the theoretical limit (around 4.9 cm for a Hamming window), although in the filtered signal used for estimating the FMCW coefficients there are four other targets close to the reference response. This result is in keeping with the capability of the HAF estimation method to extract the maximum amplitude PPS component if the ratio to the other components is above a certain threshold (around 10 dB). Besides the dramatically enhanced resolution, the correction algorithm improves the peak level of each target which leads to an increase of the signal to noise ratio with over 20 dB.

IV. MEASUREMENT RESULTS

This section is divided in two parts: the first part validates the wideband nonlinearity correction algorithm on real data and the second one is concerned with displacement measurements using the interferometric phase derived from range profiles and SAR images.

A. Nonlinearity Correction Algorithm Validation

The nonlinearity correction algorithm was first tested using only the transceiver of the implemented radar system and having artificial targets obtained with delay lines and attenuators. The tuning voltage versus frequency calibration curve of the VCO was measured. The range profiles obtained with a predistorted command signal based on the calibration curve were considered as reference. A few data sets were collected under the same external conditions as for the calibration curve measurement. Two delay lines having air-equivalent lengths of 30 cm (short path) and respectively 240 cm (long path) were used as targets. The chirp bandwidth was 4 GHz and the sweep interval 100 ms. The correction was done using both lines in order to analyze the influence of the reference response delay on the algorithm's performance. For a 4th order polynomial approximation, the HAFs plots of the FMCW coefficients in both cases are shown in Fig. 5. The estimated coefficients were used to generate nonlinearity corrected beat signals by employing the time resampling procedure. The range profiles were obtained by applying a Hamming window before the Fourier transform.

Fig. 6 shows a comparison between the range profiles obtained for the two delay lines in different cases and the range profile obtained for the predistorted sweep (in each comparison the range profiles are normalized such that the short path peak responses have the same level). The range Fig. 5. High-order ambiguity functions of the FMCW signal for two delay lines. The continuous plots show the functions for a 30 cm air-equivalent length delay line (short path) and the dotted ones for a 240 cm air-equivalent delay line (long path).

profile for a linear sweep is presented in Fig. 6a. The energy of both targets is highly spread in frequency and the main lobe for the long path occupies more than 40 resolution cells (expected in view of the high degree of nonlinearity of the VCO). The range profile corrected using the short path coefficients (shown in Fig. 6b) is similar to the predistorted sweep range profile for the short path response, but the energy of the long path is still spread and there are two main lobes for the same target. This effect is linked with the HAFs for the high-order nonlinearity terms (3 and 4) in the short reference path case which are hardly noticeable and can't be estimated properly. However, the long path calibration range profile from Fig. 6c is very similar to the one obtained with the predistorted sweep, so this software nonlinearity correction method provides good results if the calibration path is long enough to emphasize the nonlinearities (the higher order terms to be highlighted and properly estimated). A clear advantage of the proposed correction algorithm compared to the predistorted sweep technique is that the FMCW signal coefficients used for correction can be computed for each sweep and can include various frequency drifts (due to temperature, frequency pushing, etc.). The results of the range profiles comparison are summarized in Table II where the -10 dB resolutions are computed for both targets. Although the resolution for the long path corrected range profile is better than for the predistorted sweep range profile there are still present some residual nonlinearities (deterministic as well as random) which have small bandwidths and whose effect increases with range. However, they can be further mitigated with methods like those presented in [START_REF] Meta | Range nonlinearities correction in FMCW SAR[END_REF], [START_REF] Vossiek | Novel FMCW radar system concept with adaptive compensation of phase errors[END_REF], [START_REF] Avignon-Meseldzija | Compensation of analog imperfections in a Ka-band FMCW SAR[END_REF].

In order to validate the HAF-based nonlinearity estimation method for a multi-component response, a data set was acquired for a scene containing three main scatterers: one highly reflective metal disc and two vertical metal bars. The range profiles obtained in this case are shown in Fig. 7. The nonlinearity coefficients are computed on the 1.2-5.2 m range interval taking as reference target the metal disc. While on the initial nonlinear range profile obtained for the linear voltage sweep appears only a large continuous target, on the corrected profile the three targets are clearly highlighted. Notice that the power reflected by the metal disc is more than 10 dB higher in comparison to the other scatterers which is in agreement with the HAF method applicability threshold.

In the following part we show the results of the software nonlinearity correction algorithm applied to FMCW radar data acquired for a frequency sweep from 8 to 11 GHz. In Fig. 8 are shown the range profiles obtained before and after applying the correction algorithm for a scene containing a corner reflector (situated at 2.5 m from the radar) which was used as reference target for the nonlinearity estimation. In the corrected range profile the main lobe of the corner reflector target reaches the theoretical -3 dB resolution for a 3 GHz bandwidth and Hamming window (around 6.5 cm).

Fig. 9 presents the results obtained from applying the nonlinearity correction for a synthetic aperture image acquired by moving the FMCW radar on a 30 cm rail. The scene in the synthesized image contained some metal bars and one highlyreflective metal disc. The software nonlinearity correction was employed by resampling each line of the initial image before applying the matched filter algorithm [START_REF] Soumekh | Synthetic Aperture Radar Signal Processing with MATLAB Algorithms[END_REF] to obtain the synthetic aperture radar (SAR) image. Notice that in the image from Fig. 9a the targets can't be distinguished while in the corrected version they are clearly range focused (Fig. 9b).

B. Displacement Measurements

A set of simulations based on the FMCW platform's parameters was carried out in order to determine the expected theoretical accuracy of the displacement measurements. In each simulation were generated two noisy and nonlinearity corrected range profiles containing one and the same target with a given radar cross section. The target's distance from the radar in the two range profiles differed with 1 mm. The measured displacement was the one obtained from the phase difference of the target's complex amplitudes in the two range profiles. The error was computed as the difference between the 1 mm real value and the measured one. For each simulation the signal to noise ratio (SNR) was computed as the ratio between the target's peak response and the noise floor value (which for a range profile is the noise power computed in the equivalent noise bandwidth of the weighing window used before the Fourier transform). For each SNR value, the error's dispersion was computed over 1000 realizations of the two range profiles. The relationship between the displacement error dispersion and the SNR for targets placed at 2 m and 50 m is plotted in Fig. 10. Notice that for SNRs greater than 25 dB the displacement measurement error should stay below 0.1 mm.

The displacement measurements with the FMCW radar were made using both range profiles and synthetic aperture images for the 3 GHz bandwidth. The results are presented in the following sections.

1) Range profiles based displacements: Different targets such as metal bars and corner reflectors were placed in front of the radar at various ranges (1-6 m). The displacement measurements were conducted as follows. One target was placed at a certain distance from the radar and the range profile was computed after cumulating the received signal over 10 sweep periods. Afterwards, the target was displaced with a few millimeters using a caliper based device (with 0.02 mm accuracy) bonded to the target and another range profile was obtained. Due to the very small displacement in range, the target will most likely be in the same range bin in both measurements, but the phase from that bin differs proportionally with the displacement. If the range bin is not the same and the displacement is unambiguous (e.g. is lower than half the wavelength corresponding to the central frequency), the displacement is still obtained as a phase difference, but between the phases of the corresponding range bins from Normalized power [dB] no correction software corrected Fig. 8. Range profile for an indoor scene containing a corner reflector (at 2.5 m) used as reference target for the software nonlinearity correction method. The chirp bandwidth was 3 GHz and the sweep interval 100 ms. the two acquired range profiles. Anyway, the displacement is evaluated using the interferometric phase φ of the FMCW complex range profile:

δr = c 0 4πf c φ, ( 20
)
where δr is the displacement, f c the central frequency and c 0 the speed of light in air. The measured displacements and their corresponding absolute errors in a few measurements are summarized in Fig. 11. The displacements errors are below 0.1 mm which is in keeping with the SNR of over 30 dB (as resulted from the range profiles).

2) SAR Images based displacements:

The measurement procedure for the SAR images based displacements was similar to the one described in the previous subsection, but the experimental setup was the one used to obtain the nonlinearity corrected SAR image from Fig. 9b. In turn, one of the targets was moved on different directions between image acquisitions and the displacement was projected on the local line of sight (LOS) direction. Due to the short range the projection angle was different from one target to another.

The complex correlation coefficient c = ρ exp (jφ) was computed for each pair of SAR images as [START_REF] Touzi | Coherence estimation for SAR imagery[END_REF]:

c = L i=1 z 1i z * 2i L i=1 |z 1i | 2 L i=1 |z 2i | 2 , ( 21
)
where L is the number of samples used for spatial averaging and z 1i , z 2i are the complex numbers (pixels) corresponding to sample i. The magnitude ρ of the correlation coefficient is the interferometric coherence and describes the phase stability within the estimation neighborhood [START_REF] Anghel | Stable scatterers detection and tracking in heterogeneous clutter by repeat-pass SAR interferometry[END_REF]. The LOS displacements between two SAR images were computed from the phase of the correlation coefficient (the interferometric phase) using [START_REF] Carrara | Spotlight Synthetic Aperture Radar: Signal Processing Algorithms[END_REF]. The coherence and interferogram computed for two SAR images with a 3x3 pixels boxcar are shown in Fig. 12. Notice that in the proximity of the metal targets the coherence is around unity and the phase has approximatively constant value. This is linked to the fact that for the considered experimental setup, there are actually a few scatterers in a resolution cell located in the vicinity of the metal targets which leads to a less perceptible clutter. Fig. 13 shows a few measured LOS displacements and the corresponding absolute errors.

V. CONCLUSIONS

In this paper was presented an X-band FMCW radar used for displacements measurement of short-range targets. The radar's transceiver is based on an intermediary frequency architecture in order to avoid the specific homodyne problems and enhance the system's sensitivity. The nonlinearity of the VCO was mitigated using a wideband nonlinearity correction algorithm that uses HAF-based estimation and time resampling of the beat signal. The improvements due to the nonlinearity correction are clearly highlighted in the range profiles and the SAR images obtained with the FMCW radar system. The radar's measurement capabilities were validated on the data sets acquired for various targets. The displacements errors were less than 0.1 mm for metallic targets placed at a few meters from the radar.

Fig. 3 .

 3 Fig. 3. Wideband nonlinearity correction algorithm based on high-order ambiguity functions and time resampling.

Fig. 4 .

 4 Fig.[START_REF] Meta | Range nonlinearities correction in FMCW SAR[END_REF]. Range profiles simulation results: (a) 0-100m range, (b) detailed 47-51m range. In the nonlinear profile five targets are completely indistinguishable. The corrected profile leads to a resolution near the theoretical limit and improves the signal to noise ratio.

Fig. 6 .

 6 Fig. 6. Range profiles for two delay lines. The profiles obtained with a linear sweep and software nonlinearity correction are compared with the profile resulted for a predistorted sweep (shown with dotted line) computed from the measured frequency-voltage calibration curve. There are three cases considered: (a) no correction, (b) short path correction, (c) long path correction.

Fig. 7 .

 7 Fig.7. Experimental range profiles for a scene containing two metal bars and one highly reflective metal disc. On the corrected range profile the three targets are clearly separated, while in the nonlinear range profile the scatterers can't be distinguished.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Results of the software nonlinearity correction applied to the FMCW image before focusing. In the SAR image affected by the VCO nonlinearity (a) the targets are almost indistinguishable, while in the corrected image (b) the targets are clearly separated.

Fig. 11 .

 11 Fig. 11. Range profile displacement measurements results.

Fig. 12 .

 12 Fig. 12. Coherence (a) and interferogram (b) of two acquired SAR images computed with a 3x3 boxcar. The coherence is almost 1 and the phase has approximatively constant value in the proximity of the metal targets (marked by the dotted ellipses).

Fig. 13 .

 13 Fig. 13. SAR Image line of sight displacement measurements results.

ACKNOWLEDGMENT

The authors would like to thank the Électricité de France (EDF) company for supporting the development of the Xband FMCW radar experimental platform. They would also like to acknowledge the useful suggestions of the anonymous reviewers.

Silviu Ciochina is a professor of the University POLITEHNICA of Bucharest, Bucharest, Romania, and head of the Telecommunications Department, Faculty of Electronics, Telecommunications and Information Technology (since 2003). His main areas of interest are digital signal processing, adaptive algorithms, wireless communication technologies. He is author of more than 100 papers published in international journals and conference proceedings, seven of them being distinguished with a paper award. Prof. Ciochina is author or coauthor of 10 books, three of them published on international book companies. He was awarded by the Romanian Academy (Traian Vuia-1981 and Gheorghe Cartianu-1999 prizes), the Education Ministry and Defense Ministry (for works in the Radar field). He has been involved in many national and international R&D projects in the areas of communications, signal processing and radar. He was the coordinator of the Romanian teams implied in the FP6 projects ATHENA (STREP-507312), ENTHRONE 1 (IP-507637), and ENTHRONE 2 (IP 038463).


C Dumat

S Staunton

Reduced adsorption of caesium on clay minerals caused by various humic substances

The e!ect of the addition of various humic substances on the adsorption of caesium on two mineral clays has been studied. All measurements were carried out in dilute suspension under controlled conditions of temperature and ionic strength. Only a small proportion of the humic substance was adsorbed on the clays ( (10%). In general, the a$nity of the clay}humic complexes for caesium was less than that of the bare clay. The decrease was greater for illite than for montmorillonite, and greatest at trace concentrations of caesium and increased with increasing concentration of each humic substance. However, no correlation was found between the amount of humic substance adsorbed and the decrease in Cs adsorption when all complexes were considered. Neither size nor the origin of the humic substances could explain the extent of the adsorption decrease. Since neither steric hindrance nor decrease in the number of adsorption sites was the driving force behind this phenomenon, it is hypothesized that the a$nity of the clay surface is modi"ed by the organic macromolecules. The highly selective frayed edge sites of illite are particularly sensitive to the adsorption of polyanions because of their proximity to anion adsorption sites. The observed decrease in Cs adsorption may contribute to the unexpectedly high bioavailability of Cs in organic soils.

Introduction

The radioisotopes of caesium, Cs and Cs, with half-lives of 30.4 and 2.1 years, respectively, have been introduced into the environment by various events, notably fallout from the atmospheric weapons testing in the 1950s and 1960s and accidents such as the one at the Chernobyl nuclear power plant in 1986. The environmental and radioecological problems posed by caesium contamination of soils are long term, and *Corresponding author: Tel.: 0033 4 99 61 23 31; fax: 0033 4 67 63 26 14; e-mail: staunton@ensam.inra.fr accurate prediction is required. Adsorption of caesium on soil constituents is one of the factors which determines the availability of radiocaesium to biological systems. It is well known that caesium is strongly adsorbed on clay minerals [START_REF] Sawhney | Selective sorption and "xation of cations by clay minerals: a review[END_REF][START_REF] Maes | Highly selective ion-exchange in clay minerals and zeolithes[END_REF][START_REF] Cornell | Adsorption of cesium on minerals: a review[END_REF]. This adsorption is highly speci"c, particularly on illitic materials which are thought to contain a small proportion of sites, frayed edge sites (FES), which have a very strong a$nity for caesium [START_REF] Brouwer | Cesium and rubidium ion equilibria in illite clay[END_REF]. In contrast, the adsorption of caesium on organic matter is non-speci"c, and depends on the cation exchange capacity (CEC) of the organic matter and the relative proportions of caesium and other cations present. Indeed other cations may be preferentially associated with organic matter in soils and sediments. The formation of complexes between Cs and organic ligands is negligible.

Nevertheless, some observations on the fate of radiocaesium in soils lead us to suspect that organic matter does play some role in the extent and strength of adsorption. It is widely observed that radiocaesium is more easily available to biological systems in soils with high organic matter contents [START_REF] Barber | In#uence of soil organic matter on the entry of Cs into plants[END_REF][START_REF] Van Bergeijk | In#uence of pH, soil type and soil organic matter content in soil-to-plant transfer of radiocesium and strontium as analyzed by a non parametric method[END_REF], and therefore, by implication, less strongly adsorbed. A simple explanation for this would be that organic soils and peats do not contain su$cient quantities of clay materials, particularly illites, to e!ectively immobilise caesium. However, recent studies have shown that even peats contain trace amounts of illite [START_REF] Hird | Total caesium-"xing potentials of acid organic soils[END_REF], which should be more than su$cient to adsorb the trace amount of caesium present even in the most highly contaminated soils. For example, an OECD report (OCDE, 1987) gives the greatest recorded contamination of Cs due to the accident at Chernobyl in a member country as 190 kBq m\ in Sweden. As little as 1 mg illite kg\ soil (assuming the cation exchange capacity of illite to be 0.1 mol kg\ and the proportion of highly selective sites to be 1%) would have su$cient capacity to adsorb this amount of Cs. It is possible that the presence of organic matter reduces the a$nity of clay minerals for Cs. Other possible explanations of the ongoing bioavailability of Cs in highly organic upland soils depend on plant physiology, and are beyond the scope of this study.

A small number of studies have suggested that the presence of humic substances does decrease the adsorption of Cs [START_REF] Bunzl | Distribution coe$cients of Cs and Sr by mixtures of clay and humic material[END_REF][START_REF] Maguire | Caesium sorption}desorption in clay}humic acid systems[END_REF][START_REF] Hsu | Study of factors dominating sorption and desorption of Cs in soil[END_REF]. We consider that the generality of this e!ect should be tested. In a previous study, we have shown that the removal of organic matter from a soil clay increases its a$nity for caesium [START_REF] Dumat | The e!ect of removal of soil organic matter and iron on the adsorption of radiocaesium[END_REF]. For the present study, the e!ect of the addition of humic acid and fulvic acid from three di!erent origins to two reference clays, a montmorillonite and an illite is studied. We have chosen to investigate humic substances because they constitute a large proportion of soil organic matter. The amount of adsorbed humic substance in the complexes studied is far less than the organic matter content of the organic upland soils in which Cs bioavailability is a particular problem. However, if soil organic matter modi"es the a$nity of minerals for Cs, only the small proportion directly associated with clay minerals could play a role.

The e!ect was studied at two initial caesium concentrations at varying ratios of clay to humic substance. A short exchange period was used (2 h) to ensure that only true ion exchan(ge was measured, with no interference from either long term weathering e!ects of the humic substances on the clays or the slow adsorption of Cs characteristic of illites. The clays were chosen for their di!ering a$nity for caesium, resulting from di!erent adsorption mechanisms. Similarly, a comparison of the phenomenon at two concentrations of caesium, when di!erent adsorption sites may dominate adsorption, allows us to clarify the origin of the observed e!ect. At trace additions of Cs the adsorption on illites is dominated by highly selective sites which account for a small proportion of the cation exchange capacity. At larger Cs concentrations the a$nity may be determined by the regular exchange sites, which are external planar sites and are very similar to those of montmorillonite. This point will be considered in greater detail in the Discussion. Even the trace Cs concentration used in this study is in excess of levels found in contaminated soil, however, since the extent of the decrease in Cs adsorption increases with decreasing Cs loading, the e!ect of organic matter should be at least as great in situ.

Materials and methods

Two reference clays were investigated, a Wyoming montmorillonite (M) and an illite from Le Puy (I), France. Their principal characteristics are given in Table 1. The clay-sized fraction ( (2 m) was extracted from both minerals by sedimentation, saturated in Ca by repeated suspension in 1 M CaCl and dialysed until salt free and the concentration adjusted to 20 g dm\. The humic substances (HS) were isolated from three di!erent sources; Suwannee River, Peat and a Nordic Aquatic system. They were obtained from the International Humic Substance Society (IHSS) and used without further puri"cation. The composition of the humic substances is presented in Table 2. After dissolving the fulvic acids in water, and the humic acids in 0.1 M NaOH, the solutions were then adjusted to pH 7 with NaOH or HCl. Cs was obtained from Amersham International as an aqueous solution containing trace amounts of stable carrier Cs.

The Cs adsorption was carried out as follows. Triplicate suspensions of clay were prepared in polypropylene tubes by mixing appropriate volumes of clay suspension and solutions of Cs, humic substance and CaCl used as a background electrolyte. The concentrations of the various components was as follows: CaCl 3.33 mM (ionic strength 10\ M; clay 1 g dm\; humic substance 0, 0.05, 0.1 or 0.25 g dm\; CsCl 0 or 10\ M; Cs 1 or 2 MBq dm\, equivalent to 6.7 and 13.4 nM Cs (the higher Cs level for the systems containing illite which has a greater Cs adsorption capacity). The "nal pH of the suspensions was 7$0.3. The sodium present in the solutions of humic substance led to a Na concentration in the suspension of about 10\ M. The presence of this amount of sodium did not change the Cs adsorption properties of the clays. These suspensions were shaken in a reciprocal shaker for 2 h at 203C then centrifuged at 19 000 g for 30 min. Preliminary experiments have shown that these conditions are adequate to ensure complete separation of the aqueous and solid phases. Triplicate aliquots of the supernatant solution were taken, added to scintillation liquid and the radiation was measured in a Beckman LS 5801 counter.

The distribution coe$cient, K is de"ned as follows,

K "x ! /[Cs], (1)
where x ! is the amount of Cs adsorbed on unit mass of clay (mol kg\) and [Cs] (M) is the "nal concentration in solution. The value of K was calculated from the depletion of Cs from solution, by comparison with a blank solution to which no clay had been added.

A small proportion of humic substance is adsorbed, it is therefore not possible to determine the amount adsorbed by measuring the loss from solution. The amount of humic substance adsorbed on the clays was therefore interpolated from the C content clay}humic substance complexes obtained as follows. Suspensions containing 50 g clay and 5 g humic substance were shaken overnight then centrifuged at 19 000 g for 30 min. The supernatant solution was decanted o!, and the remaining solution removed by gently rinsing with water. The resulting plug was freeze dried and the C content measured using a Carlo Erba 1500. The amount of humic substance adsorbed was calculated from the known C content of the humic substance, supplied by the IHSS, and the C content of the bare clay obtained by the same method. The results of this analysis are given in Table 3. Table 4 gives the calculated values of x &1 /[HS] and x &1 for each clay#humic substance complex where x &1 is the amount of humic substance adsorbed per unit mass of clay and [HS] the "nal solution concentration. The adsorption isotherm of humic substances on clay is assumed to be linear [START_REF] Theng | The adsorption of C-labelled humic acid by montmorillonite[END_REF]. This assumption was con"rmed by preliminary experiments with illite and a di!erent fulvic acid and so these values were used to calculate the amount of each humic substance adsorbed when the initial solution concentration was 0.25 g dm\, which is used in Figs. 4 and5.

Results

Fig. 1 shows the e!ect of increasing amounts of each of the humic substances (HS) on the K value of illite with two initial Cs concentrations in solution. Fig. 2 gives the equivalent data obtained with montmorillonite. The standard deviation of the K values was about 5% and never exceeded 8%. The K values for the bare clays illustrate the well-known fact that there is greater adsorption on illite than on montmorillonite and that adsorption is highly concentration dependent, as indicated by the smaller value of K at the higher Cs concentration. The concentration dependence is more marked for illite than for montmorillonite. The addition of humic substance leads in general to a decrease in the value of K , with adsorption being less a!ected in the case of montmorillonite than illite. A more marked e!ect at trace levels of Cs than at the higher initial Cs concentration is observed. There is no signi"cant decrease in K with HS addition for montmorillonite when the initial Cs concentration is 10 M. Fig. 3 resumes the e!ect of the highest addition of humic substance on (a) illite and (b) montmorillonite for the two initial Cs concentrations. The value of K relative to that on the bare clay, denoted by *K , has been calculated for each clay#humic substance couple. The trend mentioned above is more clearly demonstrated, namely a greater e!ect on illite than on montmorillonite and at trace levels of Cs. There is no clear trend linking the nature and origin of the humic substances and the reduction in Cs adsorption.

Fig. 4 shows the relative K value, *K , for the highest HS loading as a function of x &1 , the amount of HS calculated to be adsorbed (as described in Materials and Methods) for both clays at trace additions of Cs. About 10 times more humic substance is adsorbed on illite than on montmorillonite, with an average of 33 and 3gkg\, respectively. The average value of *K is lower for the illite complexes (0.38$0.20) than that of the montmorillonite complexes (0.53$0.13), but the di!erence is not signi"cant. Fig. 5 relates the ratio of the relative K value of each illite complex, *K '

, to that of montmorillonite, *K + , at the maximum humic substance concentration and at trace levels of Cs to the relative amounts of humic substance adsorbed on each clay. There is a 10-fold di!erence in the relative amount of HS adsorbed, x &1' /x &1+ , and only a three-fold di!erence in the ratio of relative values of K , denoted by *K , on the two clays. There is no correlation between the two parameters, therefore the contrasting e!ect of humic substance on the a$nity of the two clays for Cs is not simply a result of di!ering amounts of humic substance adsorbed.

Discussion

Preliminary studies [START_REF] Staunton | Adsorption of radiocaesium on montmorillonite and illite: E!ect of charge compensating cation, ionic strength, concentration of potassium, caesium and fulvic acid[END_REF] indicated that the presence of humic substance decreased the adsorption of caesium on mineral clays. The data presented in Figs. 1 and 2 con"rm that this is a general phenomenon. Furthermore, the decrease induced by a given HS is generally enhanced as its concentration is increased. Were this e!ect observed for an actinide or a transition metal, the likely explanation would be that the solubility of the cation was increased by the formation of complexes in solution. This explanation is highly improbable for caesium, which , as presented in Fig. 3a to that of montmorillonite, *K + (Fig. 3b) at trace levels of Cs, as a function of the amount of each humic substance calculated to be adsorbed on illite, x &1'

, relative to that adsorbed on montmorillonite, x &1+ .

has a limited tendency to form complexes. There may be some association between caesium and the polyanion in solution because of the cation exchange capacity of the latter, which has, however, a preference for the divalent calcium, present at a much greater concentration (0.33;10\ M). In addition, the strong a$nity of clay minerals for caesium would completely outweigh any tendency to complex formation in solution.

If it is accepted that the decrease in Cs adsorption is not due to complex formation in solution, it must result from a change in the a$nity of the clay for caesium. Three hypotheses may be proposed to explain the change in a$nity. First, a steric e!ect; access to cation adsorption sites, in particular the sites of greatest a$nity for caesium, is impeded by the presence of the humic substance on the clay surface. Secondly, a decrease in the number of cation adsorption sites. Thirdly, a decrease in the average a$nity of the cation adsorption sites for caesium. Each of these hypotheses will be considered in turn.

Steric ewect of adsorbed humic substances

Some support for the hypothesis that the observed e!ect arises from steric hindrance comes from the relation between decreased adsorption and increase in humic substance added, and hence an increase in the amount adsorbed. However, only a small proportion of the humic substance is adsorbed because of the repulsion between the clay surface and the polyanion since both are negatively charged. There is no evidence of the adsorption of humic acids on internal sites of expanding clays such as montmorillonite. The coating of the clay surface by the humic substance is likely to be limited to positively charged edge sites caused by broken Al-OH groups. There is therefore no reason for the adsorption of humic substances on mineral surfaces to be related to the mineral CEC. Nor should adsorption depend on surface area, unless the same mineral with di!ering degrees of dispersion or grinding are compared. The positively charged edge sites are distinct from the electronegative frayed edge sites of illite (FES), with a very strong a$nity for caesium, but they may be in close proximity. This proximity lends some support to the steric hindrance hypothesis. However, a steric e!ect should be related not only to the amount of humic substance adsorbed but also to its size. It is well-known that humic acids are larger than fulvic acids, although the di!erence may be less than was previously thought [START_REF] Thurman | Molecular size of aquatic humic substances[END_REF][START_REF] Beckett | Determination of molecular weight distributions of fulvic and humic acids using #ow "eld-#ow fractionation[END_REF]. Therefore, if steric hindrance was an important factor in the reduction of Cs a$nity, the e!ects of humic and fulvic acids should di!er. No systematic di!erence was observed between the e!ect produced by humic acids and fulvic acids. In addition, although the decrease depended on the amount of HS adsorbed for a given substance, there is no correlation between x &1

and *K when all the data are considered. There is therefore no strong evidence to suggest that caesium adsorption is sterically hindered by the presence of humic substances on the clay surface. [START_REF] Hsu | Study of factors dominating sorption and desorption of Cs in soil[END_REF] have interpreted a reduction in caesium adsorption on clay in the presence of organic matter as a reduction in the number of adsorption sites. Such a phenomenon could not arise because of competition between cationic caesium and a polyanion. Organic matter, however, stabilizes clays aggregate formation, a greater number of clay platelets being associated (Vincente & Robert, 1981;[START_REF] Gu | The microstructure of dilute clay and humic acid suspensions revealed by freeze-fracture electron microscopy[END_REF]. Nevertheless, there is no evidence that this association leads to a decrease in the cation exchange capacity. Charge compensating cations on both internal and external adsorption sites remain exchangeable. Further, since aggregate formation involves planar surfaces more than clay edges, the strongly selective FES of illite should be less a!ected than the regular exchange sites, the latter playing an increasing role in caesium adsorption at higher caesium concentrations and on montmorillonite type clays. We observe the opposite; a greater e!ect on illite than on montmorillonite, particularly at trace additions of Cs. It therefore seems unlikely that the observed decrease in Cs adsorption is caused by a pro rata decrease in the number of adsorption sites, although a small decrease in the number of FES, with no consequence on the CEC could considerably a!ect the Cs a$nity.

Decrease in number of adsorption sites

Decrease in aznity for caesium

Both the clay minerals studied adsorb caesium selectively. For example the Gaines}Thomas selectivity coe$cients of the clays in the absence of humic substance and at trace levels of Cs are 7;10 and 4;10 M\ for illite and montmorillonite, respectively. As brie#y outlined in the Introduction, the strong a$nity of illite for trace amounts of caesium is usually attributed to the presence of highly selective frayed edge sites, which account for a tiny proportion (ca 1%) of the cation exchange capacity. As the amount of caesium adsorbed increases, these FES become saturated and an increasing proportion of less highly selective sites participate in adsorption, leading to a decrease in the value of K . If the decrease in Cs adsorption is due to a modi"cation of the a$nity of the clay surface when humic substances are present, then our observation that the e!ect is greater on illite at trace concentrations of Cs is important. This suggests that FES are in#uenced to a greater extent than are regular, planar exchange sites. Either humic substances are preferentially adsorbed on or near FES, or the a$nity of the FES is changed more than that of the planar sites where adsorption occurs via Ca bridges. As stated above, the polyanion, like anions such as phosphate and sulphate, is likely to be adsorbed on positively charged edge sites arising from broken Al-OH groups [START_REF] Stevenson | Clay-organic complexes and formation of stable aggregates[END_REF]. It is probable that the concentration of adsorbed humic substance is greater near the FES than on the planar surfaces. Unfortunately, the humic substances are too light to be observed directly by electron microscopy and so it is impossible to obtain direct evidence for this theory.

The presence of humic substance may also in#uence the a$nity of the FES for caesium. This has been suggested by Hird, Rimmer andLivens (1995, 1996). The strong a$nity of FES for caesium is thought to arise because Cs has a small hydration energy and so loses its hydration shell, forming a strong interaction with both clay platelets thereby closing the FES [START_REF] Sawhney | Selective sorption and "xation of cations by clay minerals: a review[END_REF][START_REF] Cornell | Adsorption of cesium on minerals: a review[END_REF]. The presence of a macromolecule within, or at the outer edge of, the FES would impede or limit its collapse and so decrease its a$nity for Cs.

The a$nity of montmorillonite for Cs may be enhanced by the presence of trace impurities of illite. However, it is well known that the a$nity of montmorillonite for the alkali metals increases in the order Na(K(Rb(Cs. This trend has been interpreted using the Hard and Soft Acid}Base Theory [START_REF] Xu | Hard and soft acid}base model veri"ed for monovalent cations selectivity[END_REF]. It is therefore probable that anything that modi"es the Misono softness parameter would in#uence selectivity. The free energy and the charge density of the clay surface are modi"ed by the presence of the humic substance [START_REF] Jouany | Surface free energy components of clay-synthetic humic acid complexes from contactangle measurements[END_REF]. Indeed, since the clay systems studied were initially homo-ionic in calcium, the K is related to the selectivity coe$cient, and hence to the thermodynamic exchange coe$cient and the Gibb's free energy of the exchange reaction [START_REF] Maes | Highly selective ion-exchange in clay minerals and zeolithes[END_REF]. Some of the humic substance is adsorbed as bridging complexes via calcium on the exchange complex. It is probable that the selectivity of Cs}Ca exchange di!ers from that of exchange between Cs and the Ca}HS surface complex. This may also have some e!ect on the overall a$nity of the surface for Cs. Overall the smaller e!ect on montmorillonite than on illite, and the smaller e!ect at larger concentrations of Cs for both clays, suggest that various mechanisms responsible for the decrease in a$nity of planar sites are less important than those acting on the FES.

Conclusions

Clay}humic substance complexes adsorb less caesium than do uncoated clays minerals. This decrease appears to be a general e!ect. For a given humic substance the e!ect increases with increasing humic substance adsorbed, although only a small proportion of the macromolecule is adsorbed by the clay. However, when all data are considered there is no correlation between the amount of humic substance on the clay and the decrease in Cs adsorption. No trend is observed for the nature (humic or fulvic acid) or origin (river, peat, aquatic) of the humic substances. A comparison of the relative e!ects on illite and montmorillonite at two concentrations of caesium leads to the conclusion that the phenomenon arises from a change in the a$nity of the clay surface for caesium. Highly selective caesium adsorption sites, the frayed edge sites of illite, are particularly in#uenced. Although the adsorption of caesium in soils and geological formations, and hence its mobility and availability, is largely determined by clay content and mineralogy, the presence of organic matter may led to signi"cant modi"cations.

Fig. 1 .

 1 Fig. 1. Value of the distribution coe$cient, K , on illite as a function of initial concentration of humic substance in solution. Two initial Cs concentrations in solution were studied; 13.4 nM or trace (left column) and 10 M (right column) for the fulvic and humic acids of Suwannee River, Peat and Nordic Aquatic system (from top to bottom). () Fulvic acid; (᭹) Humic acid.

Fig. 2 .

 2 Fig. 2. Value of the distribution coe$cient, K , on montmorillonite as a function of initial concentration of humic substance in solution. Two initial Cs concentrations in solution were studied; 6.7 nM or trace (left column) and 10 M (right column) for the fulvic and humic acids of Suwannee River, Peat and Nordic Aquatic system (from top to bottom). () Fulvic acid; (᭹) Humic acid.

Fig. 3 .

 3 Fig. 3. Value of K for the clay}humic complexes at the greatest addition of humic substance relative to that of the bare clay, *K , for each of the humic substances at each of the initial concentrations of Cs (trace level and 10 M) for (a) illite and (b) montmorillonite. White bars, trace Cs; Hatched bars, 10 M Cs.

Fig. 4 .

 4 Fig. 4. Relative K value, *K , (as explained for Fig. 3) for each clay}humic complex at trace levels of Cs as a function of the amount of each humic substance calculated to be adsorbed x &1 .(᭡) Illite; () Montmorillonite.

Fig. 5 .

 5 Fig. 5. Ratio of the relative K value for each illite complex denoted by *K ', as presented in Fig.3ato that of montmorillonite, *K + (Fig.3b) at trace levels of Cs, as a function of the amount of each humic substance calculated to be adsorbed on illite, x &1', relative to that adsorbed on montmorillonite, x &1+ .

Table 2

 2 Composition of the humic substances investigated, % by weight (analyses supplied by the IHSS)

	Table 1
	Principal characteristics of the minerals studied
	Clay	Illite	Montmorillonite
	CEC/mol kg\	0.3	0.94
	Speci"c surface area/m g\	90	800
	Major minerals	Illite	Montmorillonite
	Trace minerals	Feldspar and kaolinite	Quartz

Table 3

 3 Carbon content of bare clays and clay#humic substance complexes

	Sample

Acknowledgements

This study received partial "nancial support from the Agence Nationale pour la Gestion des DeH chets Radioactifs (ANDRA), France. Carbon analyses of the samples was carried out by the Laboratoire d'Analyse, Arras, INRA and characterisation of the humic substances supplied by the IHSS. The clay samples were gifts of D. Tessier, INRA, Versailles.


Jacinto Estima
email: jestima@masdar.ac.ae

Nicolas Fichaux
email: nfichaux@irena.org

Lionel Menard
email: lionel.menard@mines-paristech.fr

Hosni Ghedira
email: hghedira@masdar.ac.ae

The Global Solar and Wind Atlas: a unique Global Spatial Data Infrastructure for all renewable energy

Keywords: H.4.m [Information Systems Applications]: Miscellaneous Management, Documentation, Design, Standardization Spatial Data Infrastructure, Web GIS, Renewable Energy, Spatial Data Interaction, Open Geospatial Consortium, OGC, GEO, GEOSS

The Global Solar and Wind Atlas (GA) is a unique Global Spatial Data Infrastructure (GSDI) that aims to connect and bring together in a common platform the major databases and information sources around the world in the field of solar and wind energy. The initiative will be expanded to encompass all renewable energies by 2015, and will be the largest information source on renewable energy potentials ever created. It has been developed under the umbrella of the Global Atlas initiative led by IRENA in partnership with the Clean Energy Ministerial. The GSDI is based on open standards and open source and composed by fundamental datasets, a geospatial catalogue and a Web based GIS Interface. The GIS interface provides a way of interaction between users and the other components, including spatial data and tools. The GIS allows users to identify opportunity areas for renewable energy development by searching and loading data listed by the catalogue, overlaying information, and computing locally the technical potential of renewable energy. In this demonstration we aim to show the GA infrastructure and how users can interact with all the different modules with emphasis on the GIS interface.

INTRODUCTION

Doubling of the share of renewable energy globally and ensuring universal access to energy will require detailed planning for large investments in projects and infrastructures. The fundament of this ambitious goal is an accurate estimate of the local renewable energy potential for each resource. Renewable energy resources are often irregularly geographically distributed and therefore different renewable energy technologies are more or less suitable in different locations [START_REF] Meyer | Global Public Goods , Governance Risk , and International Energy[END_REF]. Based on renewable energy potential information, it is possible to identify the most suited technology opportunities for each location, identify priority development areas over a territory, plan the investments in projects and infrastructures over time, and ensure that the regulatory framework is adequate to create an attractive investment environment [START_REF] Irena | Implementation strategy for a global solar and wind atlas[END_REF].

A large technology and knowledge gap exists between nations having access to the necessary funding, technologies, and expertise to evaluate their national potentials, and those deprived of those elements.

The Global Renewable Energy Atlas is an initiative coordinated by the International Renewable Energy Agency (IRENA) 1 , which aims at federating the existing knowledge on renewable energy potentials and making it free and publicly accessible. The core of the initiative builds on a single freely accessible internet-based platform, called Global Atlas (GA), developed based on open source software and open standards.

Since 2012, the platform acts as a prospector for hotspots and a repository of the existing works at international, national and regional level. The initiative gathers datasets and methods from the most experienced institutes and private companies active in resource assessment. The platform was designed from the recommendations of the end-users, and was developed through a large international partnership. Thirty-nine countries declared their interest in the initiative, making it the largest program on renewable energy potentials. This level of international collaboration could only be achieved by using a decentralized data infrastructure, based on open standards.

The interaction between users and data is a key concern on the development of the GA. Users should access the information contained in the available datasets in an easy and complete way. Therefore, mechanisms of organizing, searching, discovering, viewing and exploring data are fundamental. The visualization of data from different sources with different structures was a challenge, and still is, and therefore the GIS interface was designed and developed based on recommendations provided by end-users.

After a brief introduction, the GA infrastructure is described, along with tools that are already available as Web Processing Services. The demo paper is finished with some conclusions and a brief explanation on the demonstration that will be provided.

System architecture

The GA platform is based open standards developed by the Open Geospatial Consortium (OGC) and on open source software in order to be fully interoperable. Standards such as OGC Catalog Service for the Web (CSW) [START_REF] Nebert | Catalogue Services Specification[END_REF], Web Map Service (WMS) [START_REF] Beaujardiere | Open Geospatial Consortium Inc . OpenGIS ® Web Map Service Implementation Specification[END_REF], Web Feature Service (WFS) [START_REF] Vretanos | Web Feature Service 2.0 Interface Standard[END_REF], Web Processing Service (WPS) [START_REF] Schut | Web Processing Service[END_REF] and Style Layer Descriptor (SLD) [START_REF] Lupp | Styled Layer Descriptor profile of the Web Map Service[END_REF] are implemented. As any SDI framework, the system architecture is distributed and composed by three major and fundamental components: Data, Catalogue and GIS Interface as shown in Figure 1. These components allow users to search for available datasets based on categories, metadata, theme, among other options, preview each dataset, and import and combine them into the GIS Interface component.

Figure 1 -System architecture Data: The data component includes all relevant datasets in solar and wind energy with local, regional and global coverage. Whenever it is possible those datasets remain in their original location but the framework has also the possibility to host data from partners that do not have their own infrastructure. 2): This is the key component that organizes all the datasets and metadata and also provides search and preview functionalities. It has the capability to harvest remote spatial data servers and also other catalogues and synchronize the metadata for stored datasets. The catalogue references the URL's of the GA data providers, and links to major data collection initiatives targeted at energy and environment applications. In that respect, the catalog is registered among the resources of the Group on Earth Observation (GEO) coordinating the development of GEOSS (Global Earth Observation System of Systems). The catalog content is also listed as a resource serving the United Nation Environment Program (UNEP) Live. Both initiatives are targeting energy and environment related resources in their global scope. The catalog is delivering a consequent subset of those 2 http://geocatalog.webservice-energy.org energy and environment resources thanks to the respect of international recommendation on standards and interoperability. The catalog currently contains around 500 references to solar and wind datasets and is continuously expanding.

GIS interface 3 (Error! Reference source not found.):

The GIS Interface has all the usual tools for zooming, panning, measuring, etc., and provides WPS tools hosting capacity. Basic users can easily overlay information and preform local analysis. A major innovation of the initiative is the capability to search the data catalogue from the GIS interface, integrate resultant datasets, and therefore create user-defined maps that can be stored, retrieved and shared through a map library. Users can add local or remote data servers to access and manipulate the information. This functionality allows the integration of user's own data that is not registered in the catalogue. Online video tutorials are designed to train basic and advanced users.

Beyond the GIS Interface, accessible using any supported browser, data registered in the catalogue is also available in the form of Web services allowing users to use their own GIS client application to access them. The infrastructure is therefore fully interoperable and data is freely and openly accessible. Specific intellectual property restrictions, negotiated with IRENA, can be taken into account by restricting the access or use of the data in the catalog and/or GIS tools. Some energy assessment related tools operated as WPS services are already available for demonstration in the GIS Interface and more are planned to be developed in the future. In the current version, existing tools only work with a specific subset of the available datasets as a consequence of the heterogeneity of data types supported by the Interface.

CONLUSIONS

The Global Atlas initiative has been developed to bridge the gap between nations having access to the necessary funding, technologies and expertise to evaluate their national potentials in terms of solar and wind renewable energy. The core of the initiative is a free and open GSDI hosting solar and wind resource maps supplied by a large international partnership. The GSDI has no equivalent so far, and brings an immense value to the renewable energy community by making available and accessible a large amount of information in an interactive format. It also provides powerful Web-GIS tools to perform local analyses of technical renewable energy potentials. This international collaboration was only made possible thanks to the OGC standards. It provides a unique data network dedicated to support the growth of renewable energy worldwide.

Although standards for sharing geospatial data, provided by OGC, are used, there are still some challenges regarding the visualization of data with diverse structures (e.g. different units, different attributes, etc.). Therefore, ways to integrate those datasets in common visualization tools need to be investigated. Another consequence of using data from several different sources is data quality and research is undergoing to develop ways to provide users with useful information on this regard.

DEMONSTRATION

For the demonstration we are planning to show the geospatial catalogue and the GIS interface. Regarding the geospatial catalogue we intend to show the search and preview capabilities and how the information is organized. Concerning the GIS interface, the interaction with maps, data, and other

Figure 2 -

 2 Figure 2 -Catalog OGC CSW

Figure 3 -

 3 Figure 3 -GIS Interface

http://www.irena.org

http://irena.masdar.ac.ae

http://www.dlr.de

http://www.energeo-project.eu

http://www.nrel.gov/

http://www.mines-paristech.eu/

functionalities, including the available analysis tools and the link between the catalogue and the GIS interface, will be demonstrated interactively. The infrastructure can be accessed at http://www.irena.org/globalatlas/, the geospatial catalogue through http://geocatalog.webservice-energy.org and the GIS Interface at http://irena.masdar.ac.ae.

DEVELOPMENT PARTNERSHIP

This GDSI has been developed and is operationally maintained by international institutes into an international partnership including IRENA, Masdar Institute of Science and Technology (MIST), DLR, NREL and MINES ParisTech.


est destinée au dépôt

Resolution of the non-stationary or harmonic Maxwell

equations by a discontinuous finite element method. application to an e.m.i. (electromagnetic impulse) case.

Françoise Bourdel, Pierre-Alain Mazet, Philippe Helluy

To cite this version:

Françoise Bourdel, Pierre-Alain Mazet, Philippe Helluy. Resolution of the non-stationary or harmonic Maxwell equations by a discontinuous finite element method. application to an e.m.i. (electromagnetic impulse) case.. 10th international conference on computing methods in applied sciences and engineering on Computing methods in applied sciences and engineering, Feb 1992, France. pp.405-422. hal-00974964