Dominique Henry

Timothée Marchal

Julien Philippe

Patrick Pons

Hervé Aubert

Classification of Radar Echoes for Identification and Remote Reading of Chipless Millimeter-wave Sensors

Keywords: radar imaging, radar polarimetry, target classification, sensor identification, wireless sensor systems

This paper describes the wireless and simultaneous interrogation and identification of multiple passive (zero-power) pressure sensors in an industrial environment with a reading range of at least 4 meters. The 3D beamscanning of the scene is performed from a 24GHz FM-CW radar and for diverse electric field polarizations. The identification is performed using a knearest neighbors classification. The benefit of using the crosspolarized electric fields combined with a radar imagery technique is enlightened from the analysis of the background clutter and the simultaneous remote interrogation of several passive pressure sensors at a distance up to 17.7m. The measurement uncertainty on pressure obtained from the proposed long-range wireless technique is finally reported and discussed.

I. INTRODUCTION

D ETECTION and remote reading of sensors in electro- magnetic reflective and/or harsh environments is challenging, especially when sensors are chipless and batteryless (or zero-power). Paradoxically, these sensors might be very useful in such environments, in which integrated circuits may be damaged and human intervention is limited. Remote sensing using Surface Acoustic Wave tags, passive Radiofrequency identification (RFID) tags or radiofrequency (RF) resonators has been investigated in harsh conditions, such as, in extreme temperatures [START_REF] Pereira Da Cunha | Wireless harsh environment SAW array system for power plant application[END_REF] [START_REF] Cheng | Wireless passive temperature sensors using integrated cylindrical resonator/antenna for harsh-environment applications[END_REF] or cluttered environments [START_REF] Mercer | RFID testing and evaluation for an RF-harsh environment[END_REF]. The lifetime of such tags in extreme conditions depends mainly on their constitutive materials, but the reliability of the wireless link between tags and their reader may be significantly degraded by electromagnetic clutter (spurious radar echoes), multiple reflections (multipath) or destructive interferences (shadow regions). It may decrease the signalto-noise ratio and eventually make tags undetectable by the reader. One solution for enhancing the signal-to-noise ratio is to take advantage of the polarization diversity. Benefits of using cross-polarized electric fields have been investigated for chipless RFID sensor tags in [START_REF] Feng | Low-cost printed chipless RFID humidity sensor tag for intelligent packaging[END_REF]- [START_REF] Girbau | Passive wireless temperature sensor based on time-coded UWB chipless RFID tags[END_REF]. However, these studies targeted short reading ranges of typically few tens of centimeters. In many industrial applications, such as Structural Health Monitoring (SHM) in inaccessible areas, larger range of wireless interrogation is often required. Cross-polarized sensor tags combined with a microwave radar imagery technique allow reaching radar-to-tag distance of 50 meters in indoor D.Henry, T.Marchal, J.Philippe, P.Pons and H.Aubert are with LAAS-CNRS and Toulouse University, Toulouse, France.

Manuscript received July 21, 2020;

environments [START_REF] Henry | Long-range wireless interrogation of passive humidity sensors using van-atta cross-polarization effect and different beam scanning techniques[END_REF]. In the paper presented at International Microwave Symposium 2020 by the authors of the present study [START_REF] Henry | Long-range zero-power multi-sensing in industrial environment using polarization diversity and 3d radar imagery[END_REF], the long-range wireless multi-sensing technique in industrial environments using zero-power (passive or chipless) pressure sensors and cross-polarized electric fields was proposed. The term long-range referred to a distance of interrogation of 3 meters or more, while the term short-range referred to reading ranges of few centimeters and is usually offered by standard chipless RFID sensor tags. Moreover, the term multi-sensing referred to the simultaneous remote interrogation of multiple sensors in the scene. We investigated in [START_REF] Henry | Long-range zero-power multi-sensing in industrial environment using polarization diversity and 3d radar imagery[END_REF] the radar interrogation of several passive pressure sensors surrounded by many electromagnetic reflective structures (such as metallic pipes, grids, walls, etc.) in an industrial environment. The reader was a 24GHz FM-CW (Frequency-Modulated Continuous-Wave) radar that performed the 3D beamscanning of the scene with two orthogonal (Vertical and Horizontal) electric field polarizations in order to enhance the signal-to-noise ratio.

We report here an extension of the work reported in [START_REF] Henry | Long-range zero-power multi-sensing in industrial environment using polarization diversity and 3d radar imagery[END_REF] by investigating for the first time the identification of chipless sensors from an original classification of radar echoes. Chipless sensors, such as in RFID applications, are usually identified in the frequency and/or time domains (see e.g. [START_REF] Vena | Chipless RFID tag using hybrid coding technique[END_REF] [10] and may require the use of several transmission lines [START_REF] Jatlaoui | Original identification technique of passive EM sensors using loaded transmission delay lines[END_REF]. The original method proposed here does not require the use of multi-band resonators for identifying chipless sensors, but it relies only on the classification of sensor radar echoes for different field polarizations.

Classification of radar echoes is usually applied to meteorological radar imagery, synthetic aperture radar (SAR) imaging or through-the-wall radar imaging (TWRI). For SAR imaging or meteorological radar images, textural and polarimetric features can be analyzed for segmentation and classification purposes [START_REF] D'elia | Sar image classification through information-theoretic textural features, mrf segmentation, and object-oriented learning vector quantization[END_REF] [13], while on TWRI application, classifications can be applied to 3D radar images [START_REF] Mobasseri | 3d classification of throughthe-wall radar images using statistical object models[END_REF]. Segmentation and identification of 3D radar targets are also possible, as shown in [START_REF] Debes | Target discrimination and classification in through-the-wall radar imaging[END_REF]. In the application reported here, we use a classification algorithm (so-called the k-nearest neighbors [START_REF] Bentley | Multidimensional binary search trees used for associative searching[END_REF]) to distinguish between radar echoes of millimeter-wave chipless sensors and electromagnetic clutter. The segmentation of radar echoes is performed using isolines computed with a marching square algorithm [START_REF] Lorensen | Marching cubes: A high resolution 3d surface construction algorithm[END_REF] in 2D azimuth/elevation planes of the radar beamscanning. The proposed classification allows to better detect and mitigate the clutter, and is also able to identify different chipless pressure sensors. To illustrate the

FM-CW radar

(H) (V) (V) or (H) (1) (2) (V) (H) pressure transducer 1 L1 L2 (1) (2) (V) (H) pressure transducer 2 L1 L2 (1) (2) (V) (H) pressure transducer N L1 L2
Fig. 1. Schematic of the 3D beamscanning principle of N zero-power sensors with a FM-CW radar using polarization diversity. Pressure sensors are here passive pressure transducers connected to two cross-polarized antennas through delay lines.

performance of the proposed classification technique applied to sensor radar echoes for very long reading ranges (8m and more), an additional beamscanning is performed to detect, identify and read two chipless pressure sensors which are located respectively at 8.1m and 17.7m in front of the radar reader.

This article is organized as follows: Section II presents the main principle of the millimeter-wave radar multi-sensing technique used in this study. Some terms used throughout this paper are reminded, such as the so-called polarization configuration (Section II-A). The identification and reading algorithm is detailed in Section II-B, and Section III describes the material and methods. Scattering parameters of three pressure chipless sensors are discussed in Section III-A while the technical characteristics of the radar reader are detailed in Section III-B. Two measurement scenarii are described in Section IV-A and the method to estimate the maximal full-scale range of the pressure measurement at each sensor position is reported in Section IV-B. Descriptors are defined in Section IV-C in order to characterize the electromagnetic clutter for the two scenarii and polarization configurations. In Section IV-D, it is shown that the clutter is dramatically mitigated from the proposed analysis of segmented radar echoes. Classifications of radar echoes for different scenarii are then performed and analyzed in Section IV-E. The pressure applied on multiple chipless sensors is estimated in Section IV-F and finally the measurement uncertainty is discussed in Section IV-G.

II. MEASUREMENT PRINCIPLE

A. Multi-Sensing and Polarization Configuration

The measurement principle is illustrated on Fig. 1. The reader unit is a FM-CW radar with carrier frequency f c and bandwidth B. The radar front-end is composed of the transmission channel (T x ) and two reception channels (R x,1 and R x,2 ). The T x antenna has narrow beamwidth in azimuth and elevation, denoted respectively by ϕ A and θ A , and its polarization is either vertical (V ) or horizontal (H). The R x,1 antenna is vertically polarized, while the polarization of the R x,2 antenna is horizontal. The radar performs the beamscanning of the scene, in which are distributed N zero power pressure (or batteryless) sensors at respective positions (ϕ 1 ,θ 1 ,R 1 ), (ϕ 2 ,θ 2 ,R 2 ),...,(ϕ N ,θ N ,R N ) in the elevation/azimuth coordinate system, where the origin (0,0,0) is the FM-CW radar location. The triplet (ϕ k ,θ k ,R k ) designates the azimuth, elevation and range from the radar of the k th pressure chipless sensor. As illustrated in Fig. 1, each sensor is composed of : (i) the zero-power pressure transducer, that is, a two-port passive network which converts the pressure variation applied to the sensor into the modification of its scattering parameters S 11 , S 12 , S 21 , S 22 at the radar operating frequency; (ii) Vand H-polarized antennas; (iii) Two delay lines of electrical lengths L 1 and L 2 , which are used for connecting the Vand H-polarized antennas to the input port (1) and output port (2) of the transducer. The analysis of the so-called backscattering antenna (or sensing) mode [START_REF] Harrington | Theory of loaded scatterers[END_REF] of such passive and wireless sensors is used for remotely estimating the pressure variation applied to the sensor. This electromagnetic backscattering originates in the impedance mismatch at input and output ports of the transducer: as the mismatch level depends on the pressure applied on the transducer, pressure variation is expected to change the radar echo level of the sensor. The electric field transmitted by the radar is here either Hor V -polarized, while the sensing mode is dual-polarized, that is, the electric field backscattered by the sensor is the combination of H-polarized and V -polarized electric fields. When the electric field transmitted by the radar is, say, H-polarized, the H-polarized component of the backscattered electric field is due to the sensing and structural modes of the H-polarized antenna of the sensor. Moreover, the V -polarized component of the field backscattered by the sensor is generated by the radiation from the V -polarized sensor antenna of the electric field received by the H-polarized sensor antenna and transmitted through the transducer and delay lines. The resulting V -polarized component of the backscattered electric field is next received by the R x,1 -antenna.

V -pol p = V V ; R k + L1 p = V H ; R k + L 1 +L 2 2 H-pol p = HV ; R k + L 2 +L 1 2 p = HH ; R k + L2
In order to clarify the various polarization configurations studied in this paper, the so-called polarization configuration, denoted by p, is defined here as follows: when the electric field transmitted by the radar is H-polarized (resp. V -polarized), then : (i) p = HV (resp. p = V V ) if the electric field is received by the R x,1 -antenna ; (ii) p = HH (resp. p = V H) if the electric field is received by the R x,2 -antenna. Throughout this paper, p = V V or HH refers to the copolarization (co-pol) configuration, while p = V H or HV refers to the cross-polarization (cross-pol) configuration. The eventual electromagnetic coupling between the two sensors antennas is assumed to be negligible. As reported in Table I, it is expected that the distance between the radar and the backscattering sensing mode depends on the chosen polarization configuration.

B. Identification and Remote Sensing Algorithm

We propose an algorithm for both the detection of radar echoes associated with the chipless pressure sensors distributed in the beamscanned scene and the estimation of the pressure at the multiple sensor locations. This algorithm is composed of the following steps (see Fig. 2): (i) Raw data is collected during the 3D mechanical radar beamscanning and for the four possible polarization configurations (p=V V , V H, HV and HH). Raw data is a 2D time data array. The first dimension is called the fast time, while the second dimension is called the slow time. The fast time is determined by the up-ramp duration (T up =20ms) of the radar triangular modulation frequency. N S (=1024) denotes the number of samples for the fast time signal. The slow time is obtained during the 3D mechanical beamscanning. The repetition time between each fast time measurement is denoted T rep and is set here to 50ms. The direction of interrogation (ϕ m ,θ q ) is associated with each fast time signal n ; (ii) The Fast Fourier Transform (F F T ) is applied over the fast time dimension to obtain the so-called beat frequency spectrum. Positive frequencies are selected and turned into range bins of theoretical resolution d (=7.5cm) ; (iii) The three-dimensional data is reconstructed for the four polarization configurations. The first dimension is the interrogation range R. The second and third dimensions are respectively the azimuth θ and the elevation ϕ ; (iv) The mean patch filter of 3×3 px 2 is applied on each elevation/azimuth plane. Then radar echoes are segmented for each elevation/azimuth plane (θ k ,ϕ k ). The segmentation is based on the marching squares algorithm [START_REF] Lorensen | Marching cubes: A high resolution 3d surface construction algorithm[END_REF] that generates contours called isolines. Isolines refer to lines along which the radar echo level does not change [START_REF] Henry | Isolines in 3d radar images for remote sensing applications[END_REF]. The echo level for which radar echoes are segmented depends on specific parameters, such as, the maximal number of local maxima inside the isoline, the initial echo level for which isolines are computed and the minimal and maximal areas inside the isoline. The following parameters are used to compute the isolines: initial echo level is -80dB, the maximal number of local maxima per isolines is 1, and the area inside an isoline ranges from 3 px 2 to 20 px 2 . A more detailed description of the algorithm can be found in [START_REF] Henry | Proximal radar sensors for precision viticulture[END_REF].The segmentation algorithm is not applied to the two copolarization configurations p=V V and HH because the lower SNR in these configurations may impact the segmentation efficiency ; (v) Isolines are indexed with some of their features, such as their coordinates, shape or echo levels inside their boundaries. We denote by C p i the i th isoline in polarization configuration p. As it will be justified in Section IV-D, only cross-polarization configurations p=V H and HV are considered in this step ; (vi) To identify the radar echoes from the pressure sensors, only isolines C V H i and C HV j that intersect in the plane (θ k ,ϕ k ) are selected. This selection is explained for two reasons: on one hand, sensing modes for p=V H and p=HV are located at the same distance R k , and on the other hand, antennas of chipless pressure sensors are located in close directions ; (vii) The remaining couples of isolines are used to distinguish between radar echoes of millimeter-wave chipless sensors and electromagnetic background clutter. The standard nearest neighbors classification [START_REF]Nearest neighbors -scikit-learn 0.23.1 documentation[END_REF] is applied to magnitude parameters derived from isolines. The performance of the classification, which depends on the environment, will be discussed in Section IV-E ; (viii) Once the couple of isolines associated with each chipless sensor is selected, we define the estimator e p max as the maximal value of echo level inside the isoline in polarization configuration p. This estimator varies with the pressure applied to the sensor. Cross-polarization configurations will be preferred in this study for the pressure estimation as it will be justified in Section IV-G.

III. MATERIAL AND METHODS

A. Passive Pressure Transducers

We consider three chipless millimeter-wave pressure sensors. They are referred as sensor 1, sensor 2 and sensor 3. These sensors are two-port wireless and passive (see [START_REF] Philippe | In-situ wireless pressure measurement using zero-power packaged microwave sensors[END_REF] for a detailed presentation). Their respective scattering parameters at 23.8GHz are given in Fig. 3. For a specified interval of applied overpressures, the full-scale measurement range of radar echoes depends on the following sensor descriptors :

• the interval ∆ Sq (where q=11, 12, 21 or 22) in which the magnitude of scattering parameter S q lies ; • the smallest magnitude S q,min (where q=11, 12, 21 or 22) of the scattering parameter S q . If ∆ Sq is high, a large measurement range of radar echo level may be expected. Moreover, very small echo levels above the noise level can be measured for low S q,min . The lowest achievable S q,min depends on the radar sensitivity (i.e. the smallest echo level detectable by the radar) and on the electromagnetic clutter. A cluttered environment may actually cause low signal-to-noise ratio and small full-scale measurement range of radar echoes. The measurement results of descriptors ∆ Sq and S q,min are reported in Table II. As it will be discussed in Section IV-F, they will significantly impact the remote sensing performances of the millimeterwave reader.

Identical cross-polarized rectangular horn antennas (gain of 20dBi) are connected at input and output ports of the sensors. Since L 1 =L 2 it can be derived from Table I that sensing modes are located at the same distance from the reader in all polarization configurations p. For sensor 1 and sensor 3, L 1 =L 2 =0.36m while for sensor 2, L 1 =L 2 =1.2m.

B. The Reader

The reader is the FM-CW radar (from IMST [START_REF]IMST Radar small and flexible radar modules by IMST[END_REF]) with carrier frequency f c =23.8GHz and modulation bandwidth B=2GHz. The theoretical depth resolution is then d= c 2B (=7.5cm), where c is the speed of light [START_REF] Piper | Receiver frequency resolution for range resolution in homodyne FMCW radar[END_REF]. The radar T x antenna is a circular horn with a dielectric lens, and its gain is of 28dBi with same beamwidth in azimuth and elevation

IV. MEASUREMENT RESULTS AND DISCUSSION

A. Measurement Scenarii

The remote sensing is performed at Chatou EDF research center, France (Electricité de France). This place includes an industrial hangar with a testing loop, called Everest, dedicated to the measurement of the flow and pressure inside pipes. It contains many metallic and massive objects that generate strong electromagnetic clutter. The chipless pressure sensors are interrogated in the two configurations illustrated in Fig. 5 and called scenario 1 and scenario 2. For the scenario 1, sensor 1 and sensor 2 are placed respectively at 4.2m and 4.8m from the reader, as depicted in Fig. 5a and Fig. 5c. According to the electrical lengths of the delay lines (see Section III-A), sensing modes are located at ranges of 4.5m and 6.0m (note that in our previous contribution reported in [START_REF] Henry | Long-range zero-power multi-sensing in industrial environment using polarization diversity and 3d radar imagery[END_REF], the reading range of sensor 1 is of 3.3m, using delay lines of electrical length L 1 =L 2 =1.2m).

For the scenario 2, sensor 2 and sensor 3 are placed respectively at 8.1m and 17.7m from the FM-CW radar, as depicted in Fig 5b andFig 5d. As a result, sensing modes are located respectively at distances of 9.3m and 18.8m from the reader. For each scenario, the radar performs a 3D beamscanning of the scene for applied overpressures varying from 0 bar to 1 bar.

B. Characterization of the highest full-scale measurement range

As reported in Section IV-A, the pressure chipless sensors are placed at various locations and may offer different fullscale measurement ranges. The highest full-scale measurement range ∆ p max (in dB scale) for a specified polarization configuration p and fixed sensor location can be defined as follows:

∆ p max = |e p max,opened -e p max,through | (1)
where statistical estimator e p max,opened evaluates the highest echo level (in dB) inside isolines when the two antennas of the sensor are open-circuited, and e p max,through estimates the highest echo levels (in dB) inside isolines, when the two-port sensor is replaced by an impedance matched transmission line.

To illustrate the physical meaning of ∆ p max , we consider the sensor 2 in the scenario 2. The corresponding sensing mode is located at a distance of 9.3m from the reader. Radar echo levels at this distance are displayed in Fig. 6a for p=V V , V H, HV and HH. As expected, the highest electromagnetic backscattering is obtained either from co-polarization configurations (p=V V and p=HH) when the sensors antennas are open-circuited, or from cross-polarization configurations (p=V H and p=HV ) when the two-port sensor is replaced by an impedance-matched transmission line. The radar echoes are segmented by red isolines according to the step (iv) of the algorithm described in Section II-B. The highest echo level is given by the statistical estimator e p max computed from echoes inside the red isoline. The lowest electromagnetic backscattering is obtained either from cross-polarization configurations (p=V H and p=HV ) when the two-port sensor is replaced by an impedancematched transmission line, or from co-polarization configurations (p=V V and p=HH) when the sensor antennas are opencircuited. The highest echo level is then given by the statistical estimator e p max computed from echoes inside the blue isoline in Fig. 6. The highest full-scale measurement range ∆ p max is then derived from eq. 1 and radar echo levels inside the red and blue isolines.

The estimation of ∆ p max is performed for each sensor and each scenario. The results are reported in Table III for the four polarization configurations. For p=V H, HV and open-circuited sensor antennas, we observe that the smallest value of e p max is lower than one obtained from p=V V , HH and the transmission line between the two sensor antennas. Consequently, ∆ p max is higher for p=V H and HV . This result can be explained by higher SNR offered by the crosspolarization configuration, since the scene does not strongly depolarize the electric field in our case. We note that ∆ p max is higher in scenario 1 than in scenario 2 because the SNR decreases with the reading range. For example, the highest full-scale measurement range ∆ HV max is of 20.2dB for sensor 2 in scenario 1 (R=4.8m) while ∆ V H max is of 14.6 dB in scenario 2 (R=8.1m). Moreover, it can be also observed from Table III that, for too long reader-to-sensor distances, the detection is no more possible in co-polarization configuration, but is still achievable using cross-polarization configuration. As a matter of fact, at the position of sensor 3 (scenario 2, R=17.7m), the algorithm of Section II-B (step (iv)) does not allow the detection of radar echoes for p=V V and HH, but it detects sensor radar echoes for p=V H and HV . The highest full-scale measurement range is then found to be of at least 7dB in this polarization configuration.

C. Electromagnetic Clutter Analysis

Isolines of the beamscanned scene are computed for 3D radar images in elevation and azimuth planes for the two scenarii (see step (iv) of the algorithm in Section II-B). Computed isolines associated with the clutter of scenario 1 are displayed in Fig. 7 in black color for p = V V , V H, HV , and HH in (θ, ϕ, R) coordinate system. It can be observed that the number of isolines differs significantly between copol and cross-pol configurations: the number of isolines is actually of 722 for p=V V , 445 for p=V H, 462 for p=HV , and 716 for p=HH. This is due to the reduction of the clutter in the cross-pol radar images compared with one obtained from co-pol radar images. In red and blue colors are displayed in Fig. 7 the isolines of sensing modes of sensor 1 and sensor 2. During the beamscanning, the applied overpressures were of 0,7 bar for sensor 1, and 1,1 bars for sensor 2. The computed isolines allow estimating the location of these sensors with angular resolutions of 1 • in azimuth and elevation, and theoretical range resolution of 7.5cm. Inside the volume of the 3D scene beamscanned by the radar in a given polarization configuration, the sensor-to-clutter radar echoes ratio Λ p is computed as follows :

Λ p = N s,p N s,p + N c,p (2)
where p denotes the polarization configuration, N s,p is the number of isolines generated by the sensors and N c,p is the number of isolines generated by the clutter. In absence of clutter in the beamscanned volume, Λ p is equal to 1 and all isolines in this volume are generated only by the sensing mode. Consequently, the applied overpressure can be derived from the analysis of the isolines, as it will be shown below. However, when the clutter is very high, Λ p is close to 0 and the isolines are generated by the clutter only. As a consequence, the sensing modes are no more detectable and the overpressure cannot be remotely estimated. Therefore, Λ p characterizes the ability to identify the sensor radar echo in a given beamscanned scene and polarization configuration. If the comparison of the number of clutters is needed between two beamscanned scenes, N c,p must be normalized by the number of pixels contained in the scene (or in the scanned volume). The scanned volume in elevation/azimuth/range coordinates is defined by : Fig. 7. Isolines (in elevation/azimuth/range coordinates) derived from the radar beamscanning of scenario 1 and computed for different polarization configurations (p = V V , V H, HV , and HH). In black color are displayed isolines of the clutter, while blue and red colors indicate isolines of the sensing modes for respectively sensor 1 (for an applied overpressure of 0,7 bar) and sensor 2 (for an applied overpressure of 1,1 bars).

V scan = 1 3 (R 3 1 -R 3 0 ) × (ϕ 1 -ϕ 0 ) × (sin(θ 1 ) -sin(θ 0 )) (3)
where R 1 and R 0 denote the highest and smallest reading ranges, ϕ 1 and ϕ 0 the highest and smallest azimuth angles, θ 1 and θ 0 highest and smallest elevation angles. We derive that V scan =2826m 3 for scenario 1 and V scan =2261m 3 for scenario 2. The number of detected radar echoes from the clutter per unit volume is given by :

N V c,p = N c,p V scan (4)
From the computed isolines, the following sensor-to-clutter radar echoes ratios and the number of isolines generated by the clutter per unit volume are derived for the two scenarii and reported in Table IV. In scenario 1, the cross-pol ratios Λ V H and Λ HV are nearly twice times larger than the copol ratios Λ V V and Λ HH . Therefore, compared with the copol configurations, the cross-pol configurations allow detecting and identifying more easily the backscattering sensing modes in presence of strong electromagnetic clutter. Inversely in scenario 2, Λ V H and Λ HV are lower than Λ V V and Λ HH . This result can be explained from observation of the pictures of scenario 2 on Fig. 5b and Fig. 5d. Sensor 2 and sensor 3 are actually located close to a long metallic pipe with circular riveted joints. These circular shapes depolarize the electric field. This interpretation is confirmed by the highest value of N V c,p obtained for p=V H or HV in scenario 2 compared with one provided by scenario 1.

D. Clutter Filtering using Intersection of Isolines

To facilitate the classification of radar echoes and filter part of the clutter, only isolines of radar echoes in crosspolarization configurations p=V H and HV that intersect are selected, as indicated in step (vi). This selection is possible because of the close positions of the two cross-polarized antennas for each chipless sensor. Consequently, radar echoes of the sensing modes for p=V H and HV are located in close elevation and azimuth angles at a given reading range. This initial filtering is illustrated in Fig. 8 for the scenario 2 at the location of sensor 3 (R=18.1m). In Fig. 8a and Fig. 8b are represented isolines for respectively p=V H (blue isolines) and p=HV (green isolines). By transposing these isolines in the same azimuth/elevation plane, we select the isolines for p=V H and p=HV that intersect. The remaining couples of isolines are shown in Fig. 8c. We note that the third green isoline for p=HV has been removed/filtered because no other isoline intersects this isoline for p=V H. We observe that the first remaining couple of isolines allows segmenting the sensing mode of sensor 3. The other one is used to segment the unfiltered clutter.

To evaluate the efficiency of this filtering, the sensor-toclutter radar echoes ratio and number of isolines generated by the clutter per unit volume, defined respectively by eqs ( 2) and ( 4), are re-computed after the step (vi) of the algorithm. These parameters are denoted by Λ (V H)∩(HV ) and N V,c,(V H)∩(HV ) . For scenario 1, Λ (V H)∩(HV ) =9.1E-3 and N V,c,(V H)∩(HV ) =7.6E-2 radar echo per m 3 . For scenario 2, Λ (V H)∩(HV ) =8.0E-3 and N V,c,(V H)∩(HV ) =1.0E-1 radar echo per m 3 . By comparing these values with the ones for p=V H and p=HV in Table IV, we observe that around 50% of the clutter has been removed by filtering in both scenarii.

E. Classification of Radar Echoes

In order to identify radar echoes of the chipless sensors, a classification of radar echoes is now proposed using the measurement results obtained from scenario 1, scenario 2 and for various overpressures applied to the sensors. The classifier used here for the identification is a standard nearest neighbors classifier [START_REF]Nearest neighbors -scikit-learn 0.23.1 documentation[END_REF]. In contrast to previously reported identification techniques (see e.g. [START_REF] Vena | Chipless RFID tag using hybrid coding technique[END_REF]- [START_REF] Jatlaoui | Original identification technique of passive EM sensors using loaded transmission delay lines[END_REF]), the proposed method does not used resonators or other passive structures for remotley identifying chipless sensors. The method proceeds only to the classification of sensor radar echoes for the four polarization configurations.

Four statistical estimators, or features, are considered here for the classification: ) transposed in the same elevation/azimuth plane for p=HH; Information about the couple of intersected isolines are referred in Section IV-D. To perform the classification, the classifier takes into consideration the 3 nearest neighbors from the couple of intersected isolines to be tested. The following classifications are here trained and tested :

• The first classification is trained with couples of isolines from scenario 1 for various applied overpressures between 0 bar and 1 bar. Classes 0, 1 and 2 designate respectively couples of isolines of the clutter, sensor 1 and sensor 2; • The second classification is trained with couples of isolines from scenario 2 for various applied overpressures between 0 bar and 1 bar. Classes 0, 2 and 3 designate respectively couples of isolines of the clutter, sensor 2 and sensor 3; • The third classification is trained with couples of isolines from both scenario 1 and scenario 2 for various applied overpressures between 0 bar and 1 bar. Classes 0, 1, 2 and 3 designate respectively couples of isolines of the clutter, sensor 1, sensor 2 and sensor 3. For each case, 50% of the beamscanning data are used to build the classes, the remaining 50% are used to test the classes. Since the four features (e V H max , e HV max , e V H max,V V and e HV max,HH ) are expected to depend on the applied overpressure, the training data includes low (0 bar) and high (1 bar) applied overpressures. Consequently, extreme values of the estimators are included in the training. In the left side of Fig. 9 is plotted e V H max,V V as a function of e HV max,HH and in the right of the figure is plotted e V H max as a function of e HV max . Data of scenario 1 are indicated by circles and data of scenario 2 are plotted using crosses. Features of the clutter, sensor 1, sensor 2 and sensor 3 are displayed respectively in black, green, blue and red colors. The features of the three sensors are distinguishable. The observed variations of the feature values for a given sensor is due to the variation of the overpressure. The difference of the feature values between each sensor originates from the different scattering parameters reported in Fig. 3. We note that e V H max,V V of sensor 2 is shifted between scenario 1 (blue circles) and scenario 2 (blue crosses). This shift is due to lower SNR provided by scenario 2. The most challenging issue is to distinguish between radar echoes of the sensors and the clutter. As observed, the number of data from the clutter is large and the features exhibit a very large variation ranging from -80dB to -40dB.

Confusion matrices of the classifications are reported in Table V. For the first classification (scenario 1), 5 beamscannings are used for the training, and 5 other for the test. Despite the large number of clutter data, the classification is easily performed. For the second classification (scenario 2), 8 beamscannings are used for the training, and 8 other for the test. All radar echoes of sensor 2 are identified. For sensor 3, the rate of identification success reaches 75%, with 6 successes, 1 failure (the radar echo is viewed as a clutter), and 1 non-detection during the step (iv) of the algorithm (this does not appear in the confusion matrix). The identification failures originate in the low SNR at the position of sensor 3 located at the distance of 17.7m. During the test of the second classification, sensor 3 is successfully identified. We observe that few spurious radar echoes due to the clutter is viewed as radar echoes of the sensors (0.04% for the training, 0.23% for the test). For the third classification using data for both scenario 1 and scenario 2, conclusions are similar to ones obtained from the previous classifications. One radar echo of sensor 2 is not identified because there is a shift of e V H max,V V between scenario 1 and scenario 2. Moreover, the number of radar echoes from the clutter which are wrongly detected as sensor radar echoes is very small (0.09% for the training and 0.17% for the test) and as expected, there is no false identification between the three sensors because their features can be distinguishable.

F. Overpressure Estimation for Various Polarization Configurations

The overpressure applied to the sensors can be derived from the isolines which have been identified as sensor radar echoes. In Fig. 10, the statistical estimator e p max is plotted as a function of the applied overpressure for all polarization configurations and for scenario 1 and scenario 2. Horizontal lines with the letter N indicate the expected lowest echo level for each sensor and scenario calculated in Section IV-B (see Table III). When a radar echo is not detected for a given applied overpressure, the default value of e p max is set to the expected lowest echo level N .

To evaluate the performance of the overpressure estimation for a specified polarization configuration p, we define the fullscale range ∆ p of the echo level for each sensor and the sensitivity α p = ∆p dP within the overpressure interval dP. The highest α p , the more sensitive the chipless pressure sensor. To compare the efficiency of each polarization configuration, we define also the ratio ∆p ∆S q , where ∆ Sq is the full-scale range of the scattering parameters within the overpressure interval dP. These parameters are reported in Table VI and the full-scale range ∆ p is indicated in Fig. 10 when it is relevant.

Sensor 1 in scenario 1 offers a high sensitivity α V V =0.75 within the overpressure range dP=0.9 bar. We note variations of e p max for sensor 1 (scenario 1) below the expected lowest echo level N for p=V V and HH. Two reasons may explain this observation: [START_REF] Pereira Da Cunha | Wireless harsh environment SAW array system for power plant application[END_REF] the two open-circuited sensor antennas used in Section IV-B do not provide here the smallest achievable echo level, and/or (2) the scene has changed between the calibration and the overpressure measurement. We observe also that the ratio ∆HH ∆S 22 =0.30 is lower than one obtained from other polarization configurations. Despite the large full-scale range ∆ S22 of 7.5 dB, the lowest magnitude of scattering parameter S 22,min =-18.7dB is too low to ensure a high full-scale measurement range of echo level (see Table II). Moreover, sensor 2 in scenario 1 offers equivalent or higher ratio ∆p ∆S q in cross-polarization configuration than in copolarization configuration. This result is explained by higher SNR and larger ∆ Sq,min . However, the pressure estimation is not possible for p=HH because most of the radar echoes are not detected.

When the reading range increases (sensor 2 in scenario 2), the ratio ∆p ∆S q decreases in all polarization configurations. The ratio is still higher for p=V H and p=HV compared with p=V V and p=HH despite lower value of Λ p (see Table IV). The full-scale measurement range of radar echo level is less impacted in cross-polarization configuration than in co-polarization configuration when the range of interrogation increases. Inversely to scenario 1, the pressure estimation for p=HH is possible. It means that the undetected radar echoes in scenario 1 are due to the clutter.

Moreover, no pressure estimation is possible when using sensor 3 in scenario 2 because parameters ∆ Sq,min are too low for the reading range of 17.7m (see Table II). Nevertheless, highest full-scale ranges ∆ V H max of 7.2dB and ∆ V H max of 8.8dB are achievable (see Table III).

G. Measurement Uncertainty of the Radar Echo Level for all Polarization Configurations

To evaluate the measurement uncertainty on the radar echo level, the beamscanning of scenario 1 is successively performed 50 times. The mean value and standard deviation of the echo level, as well as isolines, are derived in linear scale for all (sensors and clutter) isolines. The mean value and standard deviation of e max,p are computed for each isoline and displayed in Fig. 11a in dB scale. Each circle represents a radar echo measured 50 times. Red and blue colors designate respectively the cross and co-polarization configurations. It can be observed that mean(e max,p ) and std(e max,p ) are strongly correlated, and mean(e max,p ) increases with std(e max,p ). Let define the measurement uncertainty ǫ p on the measured echo level as follows:

ǫ p =
std(e max,p ) mean(e max,p )

The measurement uncertainty ǫ p (in %) is displayed on Fig 11b as a function of mean(e max,p ). ǫ p decreases with mean(e max,p ) and as expected, the overpressure estimation derived from isolines of low radar echo level is less accurate than estimations obtained from isolines of higher echo level. However, compared with co-polarization configurations, the impact on measurement uncertainty is strongly reduced for cross-polarization configurations. The mean value of ǫ p is plotted with black dashed lines: ǫ p =13.7% for p=V V and p=HH and ǫ p =7.3% for p=V H and p=HV . As an example for sensor 2 , it can be observed from Fig 10a that e max,p varies between -60dB and -70dB. From Fig. 11b, we derived that the mean value of ǫ p is 7.7% for p=V V and HH in this range of echo level. The measurement uncertainty on the radar echo level decreases in cross-polarization configurations: V V / S11 2.2 2.9 2.5 0.75 2.8 4.2 3.1 0.66 V H / S12 0.9 2.5 1.0 0.36 2. In Fig. 11c is displayed the sensor-to-reader distance R as a function of ǫ p . In co-polarization configuration, we observe that the measurement uncertainty increases with R. This effect is not apparent in cross-polarization configuration. As an illustrative example, for the sensor-to-reader distances of 15m and 20m, the mean value of ǫ p is of 12.45% for p=V V and HH while ǫ p is lower (7.2%) for p=V H and HV .

V. CONCLUSION

We have demonstrated in this paper that cross-polarized electric fields combined with a 3D radar imagery technique can be advantageously used to perform both the identification and the long-range (> 3 meters) reading of passive and wireless sensors in cluttered environments. Using a classification algorithm, we have also demonstrated the feasibility to efficiently distinguish between sensor radar echoes and the clutter. Different chipless and passive pressure sensors have been successfully identified in a cluttered environment and remotely read by using polarimetric features. For long reading ranges of interrogation (> 8 meters), it is shown that crosspolarization configuration should definitively be preferred to the copolarization configuration.

The proposed 3D analysis of the electromagnetic clutter from isolines can be advantageously applied to avoid shadow regions in the scene when distributing the chipless millimeterwave sensors. In future work, sensors allowing the simultaneous measurement of various physical quantities, such as pressure, temperature and humidity, will be investigated from the proposed remote sensing technique. Moreover, limits of the classification method will be studied to determine the maximal number of identifiable chipless sensors.

Fig. 2 .

 2 Fig. 2. The different steps of the proposed identification and remote sensing algorithm for the detection and remote reading of chipless and passive millimeter-wave sensors distributed in a cluttered environment.

Fig. 3 .

 3 Fig. 3. Measured scattering parameters at 23.8GHz of (a) sensor 1, (b) sensor 2 and (c) sensor 3 as a function of the applied overpressure. ∆ Sq and S q,min (where q=11, 12, 21 and 22) denote respectively the full-scale range and smallest magnitude of the scattering parameter Sq for an overpressure ranging from 0 to 1 bar.

Fig. 4 .

 4 Fig. 4. FM-CW radar mounted on a pan-tilt platform. The reader performs a beamscanning in azimuth and elevation. (ϕ A =θ A =6 • ). R x,1 and R x,2 antennas are rectangular horns with a gain of 20dBi. The output power of the radar frontend is of 10dBm (10mW). As depicted in Fig 4, the radar is mounted on a pan-tilt plateform performing the mechanical beamscanning. Angular resolutions both in azimuth and elevation are of 1 • .

Fig. 5 .

 5 Fig. 5. Top views and pictures of (a)-(c) scenario 1 and (b)-(d) scenario 2. In scenario 1, sensor 1 and sensor 2 are located at 4.2m and 4.8m from the reader. In scenario 2, sensor 2 and sensor 3 are respectively located at 8.1m and 17.7m from the reader.

Fig. 6 .

 6 Fig. 6. Isolines generated in the azimuth/elevation planes at the range R=9.3m at the location of the sensor 2 (scenario 2) when the two-port sensor is replaced by: (a) an open-circuit at each input port of the two sensor antennas, and (b) by an impedance-matched transmission line. Dark to bright colors in the figures indicate low (-90dB) to high (-60dB) radar echo levels.

  values are in dB scale o: open-circuited two-port -: matched-impedance transmission line ND: radar echo not detected

Fig. 8 .

 8 Fig. 8. Isolines generated in the azimuth/elevation planes at the range R=18.1m at the location of the sensor 3 (scenario 2) for (a) p=V H and (b) p=HV . Dark to bright colors indicate low (-90dB) to high (-60dB) echo levels. (c) Only couples of isolines that intersect are selected.

Fig. 9 .

 9 Fig. 9. (a) e HV max,HH as a function of e V H max,V V and (b) e HV max as a function of e V H max of the data used to train the classification. Data of scenario 1 are represented with circles. Data of scenario 2 are represented with crosses. Features of the clutter, sensor 1, sensor 2 and sensor 3 are represented respectively in black, green, blue and red colors.TABLE V CONFUSION MATRICES OF TESTED AND TRAINED DATA FOR THE DIFFERENT SCENARII trained predicted tested predicted 0 1 2 0 1 2

Fig. 10 .

 10 Fig. 10. Statistical estimator e p max as a function of the applied overpressure for p=V V , V H, HV and HH: (a) scenario 1 with sensor 1 (green crosses) and sensor 2 (blue triangles) and (b) scenario 2 with sensor 2 (blue crosses) and sensor 3 (red triangles). Horizontal lines labeled with the letter N represent the expected lowest echo level for each sensor and scenario.

  3 2.7 2.5 0.85 HV / S21 1.3 2.5 1.5 0.52 3.4 2.8 3.8 1.21 HH / S22 2.7 9.0 3.1 0.30 NR a 6.9 NR a NR a S11 2.0 4.6 1.6 0.43 NR a 8.0 NR a NR a V H / S12 1.8 3.1 1.4 0.58 NR a 1.8 NR a NR a HV / S21 2.1 3.2 2.1 0.65 NR a 1.9 NR a NR a HH / S22 3.5 7.5 2.7 0.46 NR a 1.6 NR a NR a dP=1.3bars dP=0.8bar scenario 2 ∆p in dB; ∆S q in dB; αp in dB per bar; ∆p ∆ Sq in dB per dB a NR: not relevant the mean value of ǫ p is 5.8% for p=V H and HV for -70dB≤ e max,p <-60dB.

Fig. 11 .

 11 Fig. 11. Mean value mean(emax,p) of the statistical estimator emax,p as a function of (a) standard deviation std(emax,p) and (b) the measurement uncertainty ǫp defined in Eq. (5). (c) Sensor-to-reader distance in function of ǫp. Isolines are computed from 50 3D radar beamscannings of scenario 1. Red and blue circles indicate isolines obtained from respectively co-polarization and cross-polarization configurations.

TABLE II DESCRIPTORS

 II ∆ Sq AND S q,min OF THE THREE CHIPLESS PRESSURE SENSORS AND FOR APPLIED OVERPRESSURE RANGING FROM 0 BAR TO 1

				BAR.
	q	sensor 1 ∆S q Sq,min	sensor 2 ∆S q Sq,min	sensor 3 ∆S q Sq,min
	11	3.2	-10.0	4.6	-13.4	14.3	-32.5
	12	2.8	-8.2	3.1	-11.6	2.4	-9.3
	21	2.8	-8.0	3.2	-11.6	2.5	-10.0
	22	9.5	-18.7	7.5	-16.4	2.1	-6.0
	all values are in dB

TABLE III HIGHEST

 III FULL-SCALE MEASUREMENT RANGE ∆ p max FOR ALL SENSOR POSITIONS AND POLARIZATION CONFIGURATIONS.

TABLE IV SENSOR

 IV -TO-CLUTTER RADAR ECHOES RATIO Λp AND NUMBER OF ISOLINES GENERATED BY THE CLUTTER PER UNIT VOLUME N V,c,p FOR THE TWO SCENARII IN ALL POLARIZATION CONFIGURATIONS.

			scenario 1	scenario 2
	p	Λp	NV,c,p a	Λp	NV,c,p a
	V V	2.8E-3	2.5E-1	11.6E-3	0.7E-1
	V H 4.5E-3	1.5E-1	4.4E-3	2.0E-1
	HV 4.3E-3	1.6E-1	4.5E-3	1.9E-1
	HH 2.8E-3	2.5E-1	13.2E-3	0.6E-1
	Ns,p = 2

a unit in radar echoes per m 3

TABLE VI PERFORMANCE

 VI OF THE CHIPLESS PRESSURE SENSORS FOR ALL

		POLARIZATION CONFIGURATIONS AND SCENARII
	p / Sq	sensor 1 ∆p ∆S q αp	∆p ∆ Sq	sensor 2 ∆p ∆S q αp	∆p ∆ Sq

ACKNOWLEDGMENT

The authors wish to thank Occitanie Regional Council (CARANUC Project) and Electricité de France (EDF) company for financial support. This work was partly supported by LAAS-CNRS micro and nanotechnologies platform members of the French RENATECH network.


Danièle Hauser

Céline Tison

Jean-Michel Lefèvre

Juliette Lambin

Thierry Amiot

Lotfi Aouf

Fabrice Collard

Patrick Castillan

Hauser Danièle

Tison Céline

Lefèvre Jean-Michel

Lambin Juliette

Collard Fabrice

Castillan Patrick

Measuring ocean waves from space: objectives and characteristics of the China-France Oceanography SAtellite (CFOSAT)

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

ABSTRACT

The Chinese and French Space Agencies are jointly preparing a satellite mission devoted to the monitoring of the ocean surface and related science and applications. This is the so-called "China France Oceanography SATellite" (CFOSAT), to be launched around 2013. This mission will provide simultaneous and collocated observations of wind at the ocean surface and spectral properties of surface ocean waves using two scatterometers, both in Ku-Band: SWIM for measurements of directional wave spectra and SCAT for wind vector measurements. The SWIM instrument will use a real aperture observation technique so as to avoid limitations encountered with SAR systems. This paper describes the main objectives and characteristics of the mission with a focus on the SWIM instrument designed and developed under French responsibility to measure directional spectra of ocean waves.

THE CFOSAT MISSION

Oceanography greatly benefits from remote sensing satellites for global monitoring and forecast of the sea state. The CFOSAT (China France Oceanography SATellite) mission, whose launch is planned for 2013, is designed to provide at the global scale, observations of wind at the ocean surface and spectral properties of surface ocean waves. It will serve both operational needs for the surface wind and wave forecast (marine meteorology and climatology), and research needs by improving our knowledge on waves hydrodynamics, on interactions between waves and the atmospheric or oceanic boundary layers, and on interactions between electromagnetic signals and the ocean surface.

The CFOSAT mission consists of a polar orbiting system at about 514 km altitude. The platform will embark two payloads, both are radar at Ku-band (13.2 to 13.6 GHz) scanning around the vertical axis; one is SWIM 'Surface Waves Investigation and Monitoring)' , which is a 6-beam radar at small incidence (0 to 10°), with the nadir pointing beam providing like standard altimeter missions wind speed and significant wave height and the 6 to 10° beams devoted to the measurement of directional spectra of ocean waves; it is a wave scatterometer; the other one is SCAT which is a rotating fanbeam wind-scatterometer covering the 20-60° incidence angles. This combination of small to medium incidence angles is appropriate to measure simultaneously the wind vector (from large incidence) and wave properties (from small incidence). The combination of the two instruments will help to separate the different effects governing the evolution of the wave field.

CFOSAT will provide at the global scale, observations of wind vector, directional energy spectra of ocean waves (with wavelength between 70 and 500 m), significant wave height, steepness of long waves, and parameters characterizing the statistics of wave slopes (for waves with wavelength larger than a few tens of centimetres). The mission is designed to get a global full coverage of these parameters. Near-real time transmission of the data to ground station centres, will be used to estimate geophysical parameters which will feed data assimilation into numerical models and forecast procedures.

THE SWIM INSTRUMENT

SWIM is designed to measure spectral properties of ocean waves based on the Jackson et al.'s concept [START_REF] Jackson | An analysis of short pulse and dual frequency radar techniques for measuring ocean wave spectra from satellites[END_REF] and used since the 90's on airborne systems [START_REF] Jackson | A comparison of in situ and airborne radar observations of ocean wave directionality[END_REF][START_REF] Hauser | RESSAC: A new airborne FM/CW radar ocean wave spectrometer[END_REF]. The principle is based on the analysis of backscatter modulation due to the tilt of long waves, which is maximum when the look direction is parallel to the propagation direction of the waves. Unlike observations from 2 Copyright © 2010 by ASME SAR, the modulation is integrated over the azimuth dimension of the footprint, and the discrimination in azimuth is obtained through an azimuth scanning. First ideas of designing such a system for a satellite configuration were presented by [START_REF] Jackson | Aircraft and satellite measurement of ocean wave directional spectra using scanning-beam microwave radars[END_REF], and first detailed specifications of a satellite mission was proposed by [START_REF] Hauser | SWIMSAT: A real aperture radar to measure directional spectra of ocean waves from space, Main characteristics and performance simulation[END_REF].

SWIM is a modified version of the design proposed in [START_REF] Hauser | SWIMSAT: A real aperture radar to measure directional spectra of ocean waves from space, Main characteristics and performance simulation[END_REF], now developed as part of the CFOSAT mission. SWIM will operate in Ku-band (13.575 GHz), pointing at incidence angles between 0 and 10° (multi-incidence beams) with a beam scanning and in azimuth (0-360°).

The 6 to 10° beam are designed specifically for measuring directional ocean wave spectra in the wavelength range [70, 500 m] at a scale of about 70 km x 70 km on a 180 km wide swath with a minimum detectable wavelength of about 70m and an angular resolution better than 15°.

The nadir looking beam is designed to measure wind speed and significant wave height with a concept and accuracy similar to standard altimeter missions (e.g.similar to JASON specifications).

All the beams (0 to 10°) will also be used to derive properties of the wave slope statistics from the profiles of radar cross-section.

A scheme of the geometry of observation is shown in Figure 1. For an orbit altitude of 500 km, the footprint will be about 18 km x 18 km. In the look direction, the signal is analyzed with a high range resolution. The footprint will sweep a pseudo-circle with a diameter ranging from 18 to 90 km for incidence angles ranging from 2 to 10°. The surface pattern described by the instrument for all the incidence angles is shown in Figure 2 for successive scans over 360° in azimuth, and for a satellite motion of 7 km/s. The number of elementary measurements (prior to averaging processes) is represented by the colour code. As the feed horns of the antenna will not be aligned on the antenna baseplate, switches of 90° in azimuth view angles occur between some beams, leading to this ground pattern. Nonetheless, at the scale of a resolution cell (70km along-track), all the orientations are covered.

The acquisition durations spent on each incidence angle are called cycles. The global incidence coverage [0°-10°] lasts a "macro cycle" of 218 ms.

The transmitted signal is a chirp, a linearly frequency modulated pulse, with a 320 MHz bandwidth and a 50 µs time length. After reception, the signal is compressed in range onboard, using numerical compression, allowing to reach a range resolution of about 47cm. Besides, each beam's PRF (Pulse Repetition Frequency) is selected in order to maximize the number of independent echoes, under some constraints such as the duty cycle of the high power amplification subsystem and the Tx/Rx chronogram. The pulses acquired during each cycle are averaged in time and in range in order to reduce speckle and thermal noise. For data downlink constraints, some averaging (in time and range) has to be performed on-board.

The algorithm for signal time-integration will compensate in real-time for migrations due to range variations of a surface cell from one pulse to the other due to satellite advection and antenna rotation [START_REF] Enjolras | Castillan SWIM: a state-of-the art multi-incidence beams Ku-band waves scatterometer to go beyond current radar systems[END_REF]. For the off-nadir beams (6 to 10°), the raw data transmitted to the ground stations will consist in the mean wave form computed in real-time with range migration compensation: signal intensity versus range with a range resolution of about 1.5 m integrated in time over 30 to 35 ms depending of the beam incidence. The geophysical products (directional wave spectra) will be computed at ground.

The validity of the principle to derive the spectra of ocean waves from this concept has been demonstrated several times using airborne systems developed at NASA (ROWS system, [START_REF] Jackson | A comparison of in situ and airborne radar observations of ocean wave directionality[END_REF]) and in France (RESSAC radar, [START_REF] Hauser | RESSAC: A new airborne FM/CW radar ocean wave spectrometer[END_REF]). Simulation approaches have been used first to assess the feasibility of the concept [START_REF] Hauser | SWIMSAT: A real aperture radar to measure directional spectra of ocean waves from space, Main characteristics and performance simulation[END_REF] and are presently extended and used to perform detailed analysis of the system performance [START_REF] Tison | A spaceborne radar for directional wave spectrum estimation: first performance simulations[END_REF]. The retrieval of the directional spectrum of surface waves is based on the spectral analysis of the radar backscatter modulations in each look direction, providing a modulation spectrum versus wave number of the ocean waves. After taking 3

Copyright © 2010 by ASME into account noise corrections (in particular speckle noise), and normalization factors, the modulation spectrum is related through a linear relationship to the slope spectrum of the waves, k 2 F(k,φ)) where F (k,φ) is the wave height spectrum, k the wavenumber of the waves and φ the look direction. The azimuth scanning provides the complete information (directional spectrum), with however a 180° ambiguity in the propagation direction. Overlap of the scans of the 6, 8, 10° beams will be used to estimate directional wave spectra within surface boxes of about 70 km x 70 km.

The nadir pointing mode will be used to estimate significant wave height and wind speed as in conventional altimeter missions.

The 0 to 10° radar mean cross-sections will be used to estimate statistical properties of wave slopes (short scales) as inspired by [START_REF] Chapron | Importance of peakedness in sea surface slope measurements and applications[END_REF][START_REF] Hauser | A study of the slope probability density function of the ocean waves from radar observations[END_REF]

SIMULATION OF PERFORMANCE OF WAVE SPECTRA ESTIMATES

A simulation tool, called SimuSWIM, has been developed to derive the quality estimation of waves from the SWIM characteristics. SimuSWIM is an end-to-end simulator whose main steps are: simulation of the sea surface knowing analytical wave spectrum, -computation of measured backscattered intensity, -signal inversion to estimate wave spectrum, -comparison of the estimation with the input spectrum.

Figure 3 proposes two examples of a theoretical spectrum used as input and the corresponding estimated one, taking into account the up-to-date SWIM definition. The results are accurate as they are very close to the theoretical spectrum used as input.

The modulation spectrum is a random variable. Thus, in [START_REF] Tison | A spaceborne radar for directional wave spectrum estimation: first performance simulations[END_REF], the expression of the theoretical standard deviation σ has been derived.

) ( ) ( 1 ) ( ) ( ) ( 2 ) ( ) ( 1 1 ) ( 2 2 2 2 2 2 2 ˆk R k P L N k R k P k P N k R k P N L N k b dis imp m sp imp sp imp k s p m + + = σ (1)
with P sp the speckle spectrum, P m the modulation spectrum, R the impulse response, P b the thermal noise spectrum, N imp the number of impulses averaged in time, L dis the number of samples averaged in space, N s the number of spectrum averaged, L k the number of samples averaged in wavenumber.

This enables to define the 3σ margins around the mean value. The estimation (Fig. 3) is quite entirely within these margins. Nonetheless, the simulation itself induces some artifacts due to digitalization and under-sampling. This may explain why some values are outside the +/-3σ margins.

Figure 4 illustrates the simulation on the 2D spectrum in a mixed sea case (non fully-developed wind sea spectrum and swell) compared to the reference. As automatic comparisons are not easy, some analysis tools are under development to automatically partition the spectrum into main sea states and calculate the associated geophysical parameters such as propagation direction, dominant wavelength, energy, etc.

Until now, theoretical analyses and simulations have confirmed that SWIM design is compliant with scientific requirements.

These results are based on two criteria calculated from the simulated modulation spectrum, m P ˆ: i) a wavenumber- dependent criteria, ε(k), and ii) an integrated criterion ∆Ε :

∫ = - = ∆ = max min ) , ( ˆ where 1 ) ( ) ( ) ( ˆk k m ref m P dk k k P E E E E k P k k m φ σ ε (2)
and E ref is computed with analytical shape of P m . m P ˆ is the estimated modulation spectrum. The computation of ∆E is made for a worthiest case, i.e. assuming that the estimation is equal to the reference spectrum P m plus the standard deviation:

m P m P P m + = σ . (3)
Figure 5 shows an illustration of these two criteria for the case of a wind sea modeled by a Pierson-Moskowitz function (and a wind speed of 13 m/s). Figure 5a shows that when integrated over the energy containing part of the spectrum (0.04 to 0.12 rad/m), the predicted error on the total energy is less than 15%. Figure 5b shows that for all wavenumbers within [k min ,k max ], the predicted uncertainty is less than 10%. (k min ,k max are wave numbers corresponding to the half-power energy density with respect to the peak).

Table 2 presents a summary of performances obtained for a set of simulations run in different conditions of sea-state and for an optimal set of instrument and processing parameters (parameters affecting the radar signal, m P ˆ, and its variance).

The most important parameters affecting these estimates are the averaging coefficients and the Signal to noise ratio. They are linked to the initial spatial resolution (i.e. the emitted signal bandwidth), the sensor noises and the resolution requirements. According to these simulations, the instrument and processing characteristics chosen for SWIM are appropriate to measure spectral properties of wave in a large variety of conditions.

ONGOING STUDIES RELATED TO SWIM AND CFOSAT

Several French groups are presently working on the CFOSAT/SWIM preparation.

4

Copyright © 2010 by ASME First assimilation studies have been performed at Meteo-France either using synthetic wave spectra or ENVISAT ASAR level 2 wave data [START_REF] Aouf | Recent improvements of the assimilation of upgraded ASAR Level 2 wave spectra in the wave model[END_REF]. The main conclusion is that the assimilation of directional information into wave prediction models induces a significant improvement in the estimate of wave parameters, such as the significant wave height and mean wave period. Figure 6 illustrates the impact of assimilation during the forecast period. It shows the index of assimilation, which characterizes the reduction of rms error on significant wave height gained through the assimilation, compared to Jason-1 observations. It shows that the impact of assimilation is larger (higher index of assimilation) and stays longer when directional observations are assimilated.. More generally, the assimilation system corrects well the misfit of the wave model in case of severe storm generating high swell travelling through the ocean. . Developments of swell tracking algorithms have been performed and applied on SAR wave mode of ENVISAT [START_REF] Collard | Monitoring and analysis of ocean swell fields from space: New methods for routine observations[END_REF] . They illustrate the interest of spectral wave observations to improve our knowledge of physical processes impacting swell propagation [START_REF] Collard | Persistency of ocean swell fields observed from space[END_REF][START_REF] Ardhuin | Observation of swell dissipation across oceans[END_REF]. Such approaches will also be possible for SWIM data analysis.

In order to estimate directional spectra of ocean waves using the specific sampling geometry of SWIM (see Figure 2), work is under progress to prepare level 2 products based on a the multi-beam observations. Simulations of cloud water and rain impacts on the signal are also under development to prepare rain flags.

CONCLUSION

SWIM is a new concept for measuring spectral properties of ocean waves from satellite. It is presently under development (phase B realization). It shall provide real breakthroughs in the wave observation field by providing wave spectra under large range of sea-state conditions. SWIM and SCAT combined on the same platform (CFOSAT) will foster new studies on coupled wind/wave phenomena and processes at the air/sea interface affected by wind and/or winds. The mission will be beneficial to operational forecast centers for sea-state forecast.

Figure 1 :

 1 Figure 1: Geometry of observation of the SWIM instrument

Figure 2 :

 2 Figure 2: SWIM sampling at the surface.

Figure 3 .

 3 Figure 3. Simulation of two sea states (wind sea (a) and swell (b)). The black curves are the input spectrum, the blue curves the estimated one taking into account all the properties of the instrument and of the processing chain, the red curves the speckle spectrum. The dotted lines are the +/-3σ margin around the theoretical value (σ begin the standard deviation).

Figure 4 .

 4 Figure 4. Simulated 2D wave spectra for a Jonswap spectrum (U=10 m.s -1 fetch = 60 km) after inversion (left) compared to the analytical reference (right). The resulting peak wavelength and peak direction as estimated after partitioning are noted on the figure (left top). The 180° ambiguity in direction is not removed in the example.

Figure 5 .Figure 6 :

 56 Figure 5. Quality criteria on the inverted modulation spectrum, in a wind-sea case (Pierson Moskowitz spectrum with a 13 m/s wind speed). (a) uncertainty on integrated energy (b) estimation error per wavenumber (the red curve stands for the sum all of sources of errors)


domain: info math

text:
Reza Naserasr

Sagnik Sen
email: sen@iitdh.ac.in

Éric Sopena
email: eric.sopena@u-bordeaux.fr

The homomorphism order of signed graphs

A signed graph (G, σ) is a graph G together with a mapping σ which assigns to each edge of G a sign, either positive or negative. The sign of a closed walk in (G, σ) is the product of the signs of its edges (considering multiplicity). Considering signs of closed walks as one of the key structures of signed graphs, a homomorphism of a signed graph (G, σ) to a signed graph (H, π) is defined to be a mapping that maps vertices to vertices, edges to edges, and that preserves incidences, adjacencies and signs of closed walks. This is a recently defined notion, closely related to sign-preserving homomorphisms of signed graphs (or, equivalently, to homomorphisms of 2-edge-colored graphs), that helps, in particular, to establish a stronger connection between the theories of coloring and homomorphisms of graphs and the minor theory of graphs.

When there exists a homomorphism of (G, σ) to (H, π), one may write (G, σ) → (H, π), thus extending the graph homomorphism order to a partial order on the classes of homomorphically equivalent signed graphs. In this work, studying this order, we prove that this order forms a lattice. That is to say, for each pair (G1, σ1) and (G2, σ2) of signed graphs, representing their respective classes, both their join and meet exist. While proving this result, we also show the existence of categorical products for signed graphs.

Introduction

A signed graph (G, σ) is a graph G together with a mapping σ which assigns to each edge either a positive or a negative sign. The mapping σ can be 1 given by the set E -of negative edges, in which case we may write (G, E -) instead. Given a signed graph (G, σ), the signature of G normally denotes the set of its negative edges, but it may also refer to the function σ in an equivalent way.

For the purpose of this work, graphs are finite and are allowed to have loops and multi-edges. However, multi-edges of a same sign will be irrelevant to our work and will thus not be considered. A pair of parallel edges of different signs (together with its endpoints) is referred to as a digon.

Two key notions in the study of signed graphs are the sign of a substructure, which is defined to be the product of the signs of the edges in the substructure, and switching, which is to change the sign of each edge in a given subset of edges. In most studies, including this one, these two notions are restricted to two types of substructures which are closely related: sign will be defined for cycles, or more generally for closed walks, and switching can be done on an edge cut, noting that these two structures are rather dual. They are formally defined as follows.

Definition 1 (Sign of cycles and closed walks, switching, switching equivalence). Given a signed graph (G, σ) and a closed walk W of G, the sign of W , denoted as σ(W ), is the product of the signs of the edges of W , considering multiplicities. To switch an edge cut [X, Y ] of G in (G, σ) is to change the signs of all edges in [X, Y ] and to switch a vertex subset X ⊆ V (G) is to switch the edge cut [X, V (G) \ X]. Furthermore, (G, σ ) is said to be switching equivalent to (G, σ), if (G, σ ) is a switching of (G, σ). In that case, we may also say that σ and σ are switching equivalent.

The following observations directly follow from these definitions.

Observation 2. Given a signed graph (G, σ), the sign of every closed walk of G is invariant under the switching operation. Observation 3. Given two switching equivalent signed graphs (G, σ) and (G, σ ), the symmetric difference of the two signatures σ and σ is an edge cut of G.

Switching equivalence partitions the set of all signatures on a graph G into equivalence classes, called switch-equivalence classes. Each member of such a class partitions the set of cycles (or closed walks) into two parts, the negative ones and the positive ones. The following theorem of Zaslavsky shows that this partition uniquely determines the switch-equivalence class to which a signature belongs.

Theorem 4 (Zaslavsky [START_REF] Zaslavsky | Signed graphs[END_REF]). Two signed graphs (G, σ 1 ) and (G, σ 2 ) are switching equivalent if and only if they have the same set of negative cycles.

It should be noted here that not every partition of cycles or closed walks of a graph G into two parts would correspond to a set of positive and negative cycles of a signature on G. A classification of such partitions, using the notion of exclusive 3-walk property, can be found in [START_REF] Naserasr | Homomorphisms of signed graphs: An update[END_REF] and [START_REF] Zaslavsky | Characterizations of signed graphs[END_REF].

One should also note that a signed graph (G, σ) could be simply viewed as a 2-edge-colored graph, usually associating positive edges with color blue (or with solid lines) and negative edges with color red (or with dashed lines). With such a view, all results on 2-edge-colored graphs indeed hold for signed graphs. However, the notion of switching, which is based on the basic mathematical relation between + and -(as opposed to colors red and blue which are not related), adds an intriguing level of complexity which gives birth to a number of beautiful theories.

A particular example is the notion of homomorphisms of signed graphs. While it remains strongly related to homomorphisms of 2-edge-colored graphs, it becomes of higher interest in the study of a number of conjectures, specially those in relation with the theory of graph minors, such as the four-color theorem and its possible extensions (see [START_REF] Naserasr | Homomorphisms of signed graphs[END_REF] or [START_REF] Naserasr | Homomorphisms of signed graphs: An update[END_REF] for more details on this connection).

In this paper, we study the homomorphism order of signed graphs. Because of the strong relation with homomorphisms of 2-edge-colored graphs, we will also speak about the homomorphism order of 2-edge-colored graphs, but, to emphasize the connection, and towards a uniform presentation, we will consider homomorphisms of 2-edge-colored graphs under the umbrella of edge-sign-preserving homomorphisms of signed graphs.

We continue with proper definitions of the two homomorphism relations and their connections in the next section.

Homomorphisms of signed graphs

Considering signs of cycles and closed walks as one of the key structures of a signed graph, the following is the natural definition of homomorphisms of signed graphs.

Definition 5 (Homomorphism of signed graphs [START_REF] Naserasr | Homomorphisms of signed graphs: An update[END_REF]). Given two signed graphs (G, σ) and (H, π), a homomorphism of (G, σ) to (H, π) is a mapping under which a vertex maps to a vertex, an edge maps to an edge and adjacencies, incidences and signs of closed walks are preserved. When such a homomorphism exists, we write (G, σ) → (H, π).

To present a homomorphism of (G, σ) to (H, π) as a function which not only verifies that adjacencies and incidences are preserved but also verifies that signs of closed walks are preserved, we use Theorem 4. More precisely, let φ be such a mapping. Let σ be the signature on G which is induced from (H, π) by φ, i.e., σ (e) = π(φ(e)). The condition that signs of closed walks are preserved by φ together with Theorem 4 implies that σ is switching equivalent to σ, and thus σ is obtained by switching a set X (or V -X) of vertices of G. Therefore, the mapping φ can be viewed as a function composed of three components, φ = (φ 1 , φ 2 , φ 3 ), where

φ 1 : V (G) → {+, -}, φ 2 : V (G) → V (H)
, and φ 3 : E(G) → E(H) in such a way that it preserves adjacencies, incidences and for any edge uv of (G, σ), we have σ(uv) = φ 1 (u)φ 1 (v)π(φ 3 (uv)). We note that, since switching a subset X of vertices of G is the same as switching V -X, we have

φ 1 = -φ 1 .
This view of homomorphisms of signed graphs leads to a strong connection with the notion of edge-sign-preserving homomorphisms of signed graphs, formally defined as follows.

Definition 6 (Edge-sign-preserving homomorphism of signed graphs). Given two signed graphs (G, σ) and (H, π), an edge-sign-preserving homomorphism of (G, σ) to (H, π) is a mapping φ which maps V (G) to V (H) and E(G) to E(H), in such a way that adjacencies, incidences and signs of edges are preserved. When such a homomorphism exists, we write (G, σ)

sp -→ (H, π).
As in this work we only consider signs for edges, we may simply write sign-preserving homomorphism, or in short sp-homomorphism, in place of edge-sign-preserving homomorphism.

Observe that, in a mapping of (G, σ) to (H, π), when vertices x and y of an edge e = xy of G are mapped to the two ends of a digon in (H, π), or to the same vertex which has both a negative and a positive loop, then one must indicate to which of the two possible edges e is mapped to. In all other cases, specially when H has no multi-edge, the edge mapping is induced by the vertex mapping and one may simply refer to the vertex mapping.

A key difference between the definitions of homomorphisms of signed graphs and of sp-homomorphisms of signed graphs is that in the former an edge might map to an edge of a different sign, as shown in Figure 2. However, there are also strong connections between them, that will be discussed later. For more on that point, we refer to [START_REF] Naserasr | Homomorphisms of signed graphs: An update[END_REF].

It is easily observed that both relations → and sp -→ are reflexive and associative. However, considered on the class of all signed graphs, none of them is antisymmetric. The natural way to form a partial order from either of them is to put all homomorphically equivalent signed graphs into a same class, represented by any of its members (two signed graphs (G, σ) and (H, π) are homomorphically equivalent if and only if (G, σ) → (H, π) and (H, π) → (G, σ)). Such a class will be called a hom-equivalence class. This natural partial order obtained from homomorphisms or sp-homomorphisms Figure 1: An example of a homomorphism of signed graphs of signed graphs is called the homomorphism order or sp-homomorphism order of signed graphs, respectively. The main focus of this paper is to study the homomorphism order of signed graphs.

It is a well known fact that the homomorphism order of graphs is a lattice, that is to say that any pair of graphs admits a join and a meet. The join of two given graphs is simply their disjoint union, and their meet is given by the graph product defined below. This product is known under various names such as categorical product, direct product, tensor product, Hedetniemi product, etc. We refer to [START_REF] Hell | Graphs and homomorphisms[END_REF] for more details.

Definition 7 (Direct product of graphs). Given two graphs G and H, the direct product of G and H, denoted by G × H, is the graph with vertex set V (G) × V (H) where a vertex (x, u) is adjacent to a vertex (y, v) if and only if x is adjacent to y in G and u is adjacent to v in H. This product has been naturally extended to edge-colored graphs and to directed graphs, implying the fact that the sp-homomorphism order is a lattice.

Definition 8 (Sp-direct product of signed graphs). Given two signed graphs (G, σ) and (H, π), the sp-direct product of (G, σ) and (H, π), denoted by (G, σ) sp × (H, π), is the signed graph with vertex set V (G)×V (H), where a vertex (x, u) is adjacent to a vertex (y, v) with a positive edge (respectively, with a negative edge) if and only if x is adjacent to y in (G, σ) with a positive edge (respectively, with a negative edge), and u is adjacent to v in (H, π) with a positive edge (respectively, with a negative edge).

We will show in the next section that the homomorphism order of signed graphs is also a lattice. The direct product of two signed graphs will be defined in two equivalent ways. One is constructive and easy to visualize while the other provides more intuition on why such a construction works. The later has a strong connection with sign-preserving homomorphisms, which is described below. We refer to [START_REF] Naserasr | Homomorphisms of signed graphs: An update[END_REF] and references therein for more on this connection.

One such important connection we would like to mention is based on the notion of double switching graph, which was introduced in [1] within a larger context, defined as follows.

Definition 9 (Double switching graph of a signed graph). Given a signed graph (G, σ), the double switching graph of (G, σ), denoted by DSG(G, σ), is the signed graph whose set of vertices is made of two disjoint copies V + and V -of the vertices of G, where each copy induces a copy of (G, σ) and, furthermore, where two vertices x + ∈ V + and y -∈ V -are adjacent with a positive edge (respectively, with a negative edge) if and only if the vertices x and y are adjacent in (G, σ) with a negative edge (respectively, with a positive edge).

Observe that if (G, σ ) is obtained from (G, σ) by switching a set X of vertices, then the corresponding sets X -⊂ V -and (V \ X) + ⊂ V + induce a copy of (G, σ ), with their complement inducing another copy of the same signed graph. This explains why this construction is called the double switching graph.

The following result is a special case of a general theorem given in [START_REF] Brewster | Edge-switching homomorphisms of edge-coloured graphs[END_REF].

Theorem 10. Given two signed graphs (G, σ) and

(H, π), (G, σ) → (H, π) if and only if (G, σ) sp -→ DSG(H, π).
One more important notion that we will use in the next section is the notion of the category of signed graphs. This is the category where objects are switch-equivalence classes of signed graphs and where homomorphisms play the role of morphisms from one object to another. But instead of dealing with switch-equivalence classes of signed graphs, we will consider any of its elements, taken as a representative of the whole class.

3 The homomorphism order of signed graphs Our goal in this section is to prove that the homomorphism order of signed graphs is a lattice. Observe that it is enough to show that for any pair of signed graphs (G, σ) and (H, π) (representing their respective homequivalence classes), both their join and meet exists. The join and meet are also given by two signed graphs, which are representative of their respective hom-equivalence classes. We will now define the direct sum and direct product for signed graphs and later on justify their names.

Definition 11 (Direct sum of two signed graphs). Given two signed graphs (G, σ) and (H, π), the direct sum of (G, σ) and (H, π), denoted by (G, σ) + (H, π), is the disjoint union of (G, σ) and (H, π).

x y u v

+ - + - - + - + x y u v + - + - - + - + x y u v + - + - - + - + x y u v + - + - - + - +
Figure 2: Possible products of two edges Definition 12 (Direct product of two signed graphs). Given two signed graphs (G, σ) and (H, π), the direct product of (G, σ) and (H, π), denoted by (G, σ) × (H, π), is the signed graph (F, ς) defined as follows.

1. V (F ) = {+, -} × V (G) × V (H).
2. To have an edge between (i, x, u) and (j, y, v), first of all we must have an edge xy in G and an edge uv in H. In such a case, the corresponding eight vertices (respectively, four vertices if either x = y and u = v, or two vertices if both x = y and u = v) will induce a matching of order four (respectively, two, one), all of its edges having the same sign as σ(xy). The choice of these edges then depends on whether uv has the same sign as xy or not, and is described as follows (see Figure 2).

(a) If σ(xy) = π(uv), then (i, x, u) is adjacent to (i, y, v) and (i, x, v) is adjacent to (i, y, u) for every i ∈ {+, -} (left side of Figure 2).

(b) If σ(xy) = -π(uv), then (i, x, u) is adjacent to (-i, y, v) and (i, x, v) is adjacent to (-i, y, u) for every i ∈ {+, -} (right side of Figure 2).

We will now show that (G, σ) + (H, π) and (G, σ) × (H, π) are the join and meet, respectively, of (G, σ) and (H, π) in the homomorphism order. Before going to that, let us review the definition of join and meet for the homomorphism order.

Definition 13 (Join and meet of two signed graphs). Let (G, σ) and (H, π) be two signed graphs.

The join of (G, σ) and (H, π) is a signed graph (J, µ) satisfying (G, σ) → (J, µ) and (H, π) → (J, µ) such that (G, σ), (H, π) → (C, η) implies (J, µ) → (C, η).

The meet of (G, σ) and (H, π) is a signed graph (F, ς) satisfying (F, ς) → (G, σ) and (F, ς) → (H, π) such that (C, η) → (G, σ) and (C, η) → (H, π) implies (C, η) → (F, ς).

Usually, for (undirected, directed) graphs, the standard way of showing that join and meet exist is to prove that the direct sum and the direct product are the categorical co-products and products, respectively. For signed graphs, we will do something similar. Let us first provide the definitions of co-product and product in the category of signed graphs.

Definition 14 (Categorical co-product of two signed graphs). Given two signed graphs (G, σ) and (H, π), their categorical co-product is a signed graph (J, µ) together with two (inclusion) homomorphisms i G : (G, σ) → (J, µ) and i H : (H, π) → (J, µ) such that, for any signed graph (C, η), if there exist homomorphisms f G : (G, σ) → (C, η) and f H : (H, π) → (C, η), then there exists a unique homomorphism f : (J, µ) → (C, η) satisfying

f G = i g • f and f H = i H • f .
Definition 15 (Categorical product of two signed graphs). Given two signed graphs (G, σ) and (H, π), their categorical product is a signed graph (F, η) together with two (projection) homomorphisms p G : (F, ς) → (G, σ) and i H : (F, ς) → (H, π) such that, for any signed graph (C, η), if there exist homomorphisms f G : (C, η) → (G, σ) and f H : (C, η) → (H, π), then there exists a unique homomorphism f :

(C, η) → (F, ς) satisfying p G = f • f G and p H = f • f H .
Thus from the definitions of join and categorical co-product it is clear that, if the categorical co-product of two signed graphs exists, then the join of their corresponding hom-equivalence classes also exists and is the same as the hom-equivalence class of their categorical co-product. Similarly, if the categorical product of two signed graph exists, then the meet of their corresponding hom-equivalence classes also exists and is the same as the hom-equivalence class of their categorical product.

Thus let us first show that (G, σ)+(H, π), together with the two natural inclusion homomorphisms, is actually the co-product of (G, σ) and (H, π).

Theorem 16. The signed graph (G, σ)+(H, π), along with the two natural inclusion homomorphisms, is the co-product of (G, σ) and (H, π).

Proof. First let us explicitly describe the natural inclusion maps. The homomorphisms i G : (G, σ) → (G, σ) + (H, π) and i H : (H, ς) → (G, σ) + (H, π) do not switch any vertex and map a vertex or an edge to itself. Let now (C, µ) be a signed graph admitting homomorphisms f G : (G, σ) → (C, η) and f H : (H, π) → (C, η). Consider then the homomorphism f :

(G, σ) + (H, π) → (C, η), given by f (x) = f G (x) for x ∈ V (G) ∪ E(G) and f (y) = f H (y) for y ∈ V (H) ∪ E(H). From the definition of f it follows that f G = i g • f and f H = i H • f . Moreover, the uniqueness of f is obvious.
Next we are going to show that (G, σ) × (H, π), along with the two natural projections, is the categorical product of (G, σ) and (H, π). We will explicitly describe the projections inside the proof.

Theorem 17. The signed graph (G, σ)×(H, π), along with the two natural projections, is the categorical product of (G, σ) and (H, π).

Proof. We must first present the natural projections p G and p H of (G, σ) × (H, π) onto (G, σ) and onto (H, π), respectively. The projection p G maps each vertex (i, x, v) to the vertex x of (G, σ) without switching. It is straightforward to check that p G is a homomorphism of (G, σ) × (H, π) to (G, σ). The projection p H first switches all vertices of the form (-, x, v), and then maps each vertex (i, x, v) to the vertex v of (H, π). That means, due to the switching, that an edge between two vertices of the form (i, x, u) and (j, y, v) will change sign if and only if i = j. Thus, from the definition of (G, σ) × (H, π), it follows that p H is a homomorphism of (G, σ) × (H, π) to (H, π).

We may now consider a signed graph (C, η) and assume that f G and f H are homomorphisms of (C, η) to (G, σ) and of (C, η) to (H, π), respectively. We would like to define a natural mapping f of (C, η) to the product (G, σ) × (H, π), using f G and f H . Recall that a homomorphism f of signed graphs consists of three components, f 1 , f 2 and f 3 , the first one deciding for each vertex whether it must be switched or not, the second one being a vertex mapping and the third one an edge mapping.

To define the mapping f , we choose f 1 to be identical to the first component of f G , that is to say, f 1 switches a vertex z of (C, η) if and only if f G switches z. For the component f 2 , if f H switches a vertex z of (C, η), then

f 2 maps z to (-, f G (z), f H (z)), otherwise f 2 maps z to (+, f G (z), f H (z)).
Finally, we must determine the component f 3 , that is the edge mapping. Let e be an edge of (C, η) with endpoints z and z . Let i be the sign of e after applying the possible switches given by f G at vertices z and z . Observe that there must be an edge with sign i between f (z) and f (z ). We then set f 3 (e) to be this edge. It is then straightforward to check that f is a homomorphism of (C, η) to (G, σ) × (H, π) and, furthermore, that

p G • f = f G and p H • f = f H .
The uniqueness of f is obvious.

Note that the direct product of two signed graphs (G, σ) and (H, π) has order 2 × |V (G)| × |V (H)|. We will understand the reason behind this unusual form of the product at the end of this section.

For now, we can say that both the categorical co-product and the product of two signed graphs exist. Therefore, the homomorphism order of signed graphs is a lattice.

Theorem 18. The homomorphism order of signed graphs is a lattice.

Proof. Follows directly from Theorems 16 and 17.

In the following we are going to describe yet another way of presenting the categorical product. This alternative construction has relation with the sp-categorical product of graphs. To describe this construction, we must introduce a few more concepts.

Given a signed graph (G, σ) and a vertex v of G, we denote by N + (v) (respectively, by N -(v)) the set of vertices of G that are adjacent to v with a positive edge (respectively, with a negative edge), allowing N + (v) (respectively, N -(v)) to contain v only if there is a positive loop at v (respectively, a negative loop at v). Two vertices u and v of a signed graph are said to be twins if and only if N + (v) = N + (u) and N -(v) = N -(u). On the other hand, two vertices u and v are antitwins if N + (v) = N -(u) and N -(v) = N + (u), or, equivalently, if after having switched (only) u, v and u are twins in the so-obtained switching equivalent signed graph (G, σ ). Observe that our definition of twins deals with loops as well. Indeed, if u is an antitwin of v in (G, σ) and u has, say, a positive loop, then u must be adjacent to v by a negative edge, and v must also have a positive loop.

Observe also that in DSG(G, σ), the two copies in V + and V -of any vertex x are antitwins. The converse is also true in the following sense.

Theorem 19. A signed graph (H, π) is a double switching graph if and only if its set of vertices can be partitioned into pairs of anti-twins.

Proof. One direction directly follows from the definition of a double switching graph. For the other direction, simply delete one vertex from each pair of antitwin vertices in (H, π) and let (G, σ) be the resulting signed graph. It is clear that (H, π) = DSG(G, σ). Note also that different choices of vertices will lead to different switching equivalent copies of (G, σ).

Let us now recall Definition 8 and use it along with Theorem 19 to redefine the categorical product of two signed graphs (G, σ) and (H, π) with respect to homomorphisms of signed graphs (see Definition 12), as the signed graph (F, η) whose double switching graph is the product DSG(G, σ) sp × DSG(H, π). This is well defined since we have the following result.

Theorem 20. Given two signed graphs (G, σ) and (H, π), the set of vertices of the direct product DSG(G, σ) sp × DSG(H, π) can be partitioned into pairs of antitwins.

Proof. Recall that, for each vertex x of G, there are two vertices x + and x -in DSG(G, σ). Thus, for each pair of vertices (x, u) with x ∈ V (G) and u ∈ V (H), we have four vertices (x + , u + ), (x + , u -), (x -, u + ) and (x -, u -) in DSG(G, σ) sp × DSG(H, π). Furthermore, if any vertex, say (y + , v + ), is adjacent to one of these four vertices, say (x + , u + ), with a positive edge, that would mean that xy is a positive edge of (G, σ) and uv is a positive edge of (H, π). Thus, x -y + is a negative edge of DSG(G, σ) and u -v + is a negative edge of DSG(H, π), which gives that (x -, u -) is the antitwin of (x + , u + ) in DSG(G, σ) sp × DSG(H, π). Similarly, (x -, u + ) and (x + , u -) form a pair of antitwin vertices in DSG(G, σ) sp × DSG(H, π). All other cases are similar.

In other words, the product (F, ς) of (G, σ) and (H, π) satisfies

DSG(F, ς) = DSG(G, σ) sp × DSG(H, π),
where (x + , u + ) is taken as the vertex (+, x, u) of (F, ς), which represents the pair ((x + , u + ), (x -, u -)) of antitwin vertices in DSG(G, σ) sp × DSG(H, π), and (x + , u -) is taken as the vertex (-, x, u), which represents the pair ((x -, u + ), (x + , u -)). Hence, the given signature ς is a function of σ, π, and their order in the product. However, the class of switching equivalent signatures to which ς belongs is only a function of σ and π, and is independent of the order in which we multiply the two signed graphs. That is, (G, σ)×(H, π) and (H, π)×(G, σ) are switching equivalent. Furthermore, if (G, σ ) is switching equivalent to (G, σ) and (H, π ) is switching equivalent to (H, π), then (G, σ ) × (H, π ) is switching equivalent to (F, ς).

Remarks

In this work, given two signed graphs (G, σ) and (H, π), we have defined their direct sum (G, σ) + (H, π) and their direct product (G, σ) × (H, π). We then defined the homomorphism order of signed graphs, whose elements are the hom-equivalence classes of signed graphs, each class containing all the signed graphs which are pairwise homomorphically equivalent.

Furthermore, we proved that the direct sum and the direct product of two signed graphs are, respectively, their categorical co-product and product. We also provided two ways of constructing the categorical product, having order 2 × |V (G)| × |V (H)|, one of them easier to visualize and the other intuitively clearer. After that we showed that the homomorphism order of signed graphs is a lattice, by showing that the categorical co-product and the categorical product of (G, σ) and (H, π) represent the classes of their join and of their meet, respectively, in the order.

The core of a signed graph (G, σ) is its smallest subgraph which is also a homomorphic image of (G, σ). The fact that this notion is well defined follows from the fact that any two such minimal subgraphs are necessarily switching equivalent, as shown in [START_REF] Naserasr | Homomorphisms of signed graphs[END_REF]. For example, the signed graph on the left side of Figure 2 is switching equivalent to the signed graph obtained by switching its negative edge to a positive one, and its core is (K 2 , +), the graph consisting of a positive edge (right side of the same figure). This signed graph is also switching equivalent to the signed graph obtained by switching its positive edge to a negative one, and its core is (K 2 , -), the graph consisting of a negative edge. Clearly, (K 2 , +) and (K 2 , -) are switching equivalent signed graphs.

A signed graph (G, σ) whose core is (G, σ) itself is called a core. Note that, if we restrict ourselves only to cores, then homomorphisms of signed graphs induce a partial order, equivalent to the homomorphism order. By what we have proved, given any two cores (G, σ) and (H, π), in the above mentioned order, the core of their direct sum and direct product will correspond to their join and meet, respectively. Thus, the partial order induced by homomorphisms of signed graphs on the set of cores is a lattice.

The two examples below will show that (1) the core of the meet of a given pair of signed graphs could be of smaller order than the order of the categorical product of the two given signed graphs, and (2) as a general construction of meet that works for all pairs of signed graphs, 2 × |V (G)| × |V (H)| is the smallest order one may work with.

Given an integer k = 0, let C k be a signed graph on the cycle of length |k| (i.e., C |k| ) with an assignment of signs to the edges in such a way that the sign of the resulting cycle is the same as the sign of k. That is to say, C k is a negative cycle if k < 0 and a positive cycle, otherwise. In particular, C -1 is the negative loop and C +1 is the positive loop. The meet of the two cycles C p and C q can then be determined as follows.

1. If p and q have the same parity and the same sign, then, assuming |p| ≤ |q|, C p × C q is homomorphically equivalent to C q . That is simply because C q → C p , and thus signed graphs admitting a homomorphism to both C p and C q are those admitting a homomorphism to C q .

2. If either the parity or the sign of p and q are not the same, then the meet of C p and C q is simply (K 2 , +), which is homomorphically equivalent to (K 2 , -). To see that, let us denote by g ij (G, σ), ij ∈ Z 2 2 , the smallest length of a non-trivial closed walk of type ij in (G, σ), that is to say, a non-trivial closed walk whose sign is positive if i = 0 and negative if i = 1, and whose length is even if j = 0 and odd if j = 1 (see [START_REF] Naserasr | Homomorphisms of signed graphs: An update[END_REF] for more details on this notion). If no such closed walk exists in (G, σ), we let g ij (G, σ) = ∞. Observe now that a signed graph (G, σ) which maps to C p has g ij (G, σ) = ∞ for at least two choices of ij ∈ Z 2 2 , where these two choices depend on the parity and sign of p. As the same holds for q, and since p and q cannot lead to the same choices, a signed graph (G, σ) which maps to both of them must have g ij (G, σ) = ∞ for at least three choices of ij ∈ Z 2 2 . Therefore, the only possible closed walks in (G, σ) are closed walks of type 00.

Thus, on the one hand C -1 × C +1 has 2 × |V (C -1 )| × |V (C +1 )| vertices, and on the other hand, for large values of |p| and |q| where either the parity or the sign of p and q are not the same, the core of the categorical product C p × C q , that is their meet, is of order 2.

Finally, it is worth noting here that similar constructions can be given, leading to similar results, for oriented graphs with respect to pushable homomorphisms, as introduced by Klostermeyer and MacGillivray in [START_REF] Klostermeyer | Homomorphisms and oriented colorings of equivalence classes of oriented graphs[END_REF].

Acknowledgement. This work has been supported by the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39) and by the French ANR project HOSIGRA (ANR-17-CE40-0022).