S Maisonneuve

P Pagano

F Alart

Lebon

EVOLUTION OF PHASES IN A SHAPE MEMORY ALLOY BY LOCAL AND GLOBAL MINIMISATIONS OF A NON CONVEX POTENTIALS

Introduction

A Shape Memory Alloy (SMA) is a mixture of martensite and austenite or only martensite or austenite. Austenite-martensite phase transformation leads to variations of composition. This transformation can appear by mechanical or thermal action. The characteristic pseudo-elastic behaviour appears from this modification in the material over certains conditions for which observe an hysteresis loop on the stress-strain relationship. This hysteresis can be influenced by several factors : thermal effect, dissipation, structural effect, existence of geometrical defect, etc ..

Here, we use a model taking the temperature, the deformation and the volumic proportions of austenite and martensite as state variables. We adopt, here, Fremond's modelling in the non dissipative case which gives a non monotonic stress-strain relationship. Some experiments [START_REF] Chrysochoos | Couplages thermomecanique du comportement d'alliages Cu-Zn-Al et Ni-Ti[END_REF] show that during room temperature test, dissipative effects are strongly dominated by effects resulting from temperature variations of material associated to the latent heat. In our case, after elimination of the volumic proportions of martensite, the bulk energy density is a non convex function.

For a given problem, it is classical to look for the mechanical equilibrium by a global minimization of the "total" energy (potential energy). Moreover, the energy associated at each problem is not quasiconvex. Existence of absolute minima is not garanteed in all the case. It is possible to pose displacement boundary-value problems for which the total energy has no minimizer and then no absolute minimum ; the minimizing sequences are produced by displacements with fine-scale irregularities. Consequently, we introduce two relaxation procedures ( convexification and quasiconvexification) for which existence of minimizer is garanteed in order to compare in the three cases, displacement solutions and evolutions of phases. The algorithm to find global minimum is based on an augmented Lagrangian method (decomposition-coordination method).

From a mechanical point of view, there is no reason to exclude local minima as possible equilibrium states. So, the evolution of phases may be different from the loading process to the unloading one. In order to fine local minima, we write the non convex potential as a difference of two convex functions (D.C.). Then we can formulate a problem of two variational inequalities in term of displacements and stresses. In the finite dimensional case, this problem is equivalent to search solutions which locally minimize the "total" energy. By applying partial convex duality, an inf-inf problem can be formulated with respect to an appropriately defined Lagrangian function.

We present a theoretical and numerical comparison between these two points of view in focusing out the evolution of phases.

Constitutive law

Here, we adopt the Fremond modelling (Fremond, 1987) with only two variants of martensite, and we denote by ;3 1 and ;3 2 their volumic proportions. We assume that the volumic free energies of the phases (respectively the austenitic one and the two martensitic ones) have the following form involving the small strain tensor e, The phases are assumed to have the same elastic tensor K, and the same heat capacity C. The austenite-martensite phase transformation is associated with a residual stress tensor ar which is be chosen identically for the two variants. L is the latent heat of phase transformation. Fremond writes the total volumic free energy \[1 as a convex combination of the three energies: the volumic proportions are constrained to stay in the simplex C via the addition of the indicator function of this simplex :

W(e, T, f3b f3z) f3I W_(e, T) + f3zW+(e, T)

+ (1-f3I-f3z)Wo(e,T) + Tlc(f3I,f3z)
where Jc(j3 1 , ;3 2 ) is the indicator function of the convex set : 0 ::; 'Yi ::; 1 'Yl + 'Y2 ::; 1} .

We introduce the formalism of generalized standard materials and we neglect dissipation (Chrysochoos et al., 1995): we can eliminate the internal variables (( 3 1 ,(3 2 ). Then, a new potential can be defined as the minimum of three convex functions. Note that this potential is not convex:

W(e, T)

min{\ll(e,T,(Jl,fJ2), (f3bf32) E C} min{Wo(e, T), W_(e, T), W+(e, T)} With the three-well energy W, which does not depend on fJ1, fJ2, the material constitutive law is now written :

a E BeW(e(u), T) (1)
where 8 denotes the subdifferential operator of Clarke.

We consider an extension test and we represent the energy W at different temperatures in Figure 2 : low temperature T ::; T 0 , intermediate temperature To ::; T ::; Tc and high temperature T 2:: Tc. With the constitutive law (1), we etablish a classical formulation of the mechanical problem.

Mechanical problem at fixed temperature

Consider a boundary problem on a domain n of boundary r. Let f denote the density of body force applied on n, F the density of surfacic force ap- plied on the part r 0 of r' and u a displacement given on the complementary part rl.

The mechanical problem at fixed temperature consists in finding the displacement field u and the stress field a verifying the following equations In the following, the set of kinematically admissible displacement fields with finite energy of the structure will be defined by :

Dad= { v E (H 1 (n)f, v =Don f 0 } and the stress spaces H = (L 2 (n)] 6 • 4. Global minimum 4.1. THE THREE-WELL ENERGY
At first, we search for a global minimum of the three-well potential energy.

To solve this problem, we consider an augmented Lagrangian. It is a function of a multiplier JL and of two independent variables v,q corresponding to the displacement and strain fields.

Lr(v, q, J.L) = { W(q) dx -l(v) + { JL: (e(v)-q) dx + :_ { ((e(v)-q)) 2 dx ln ln 2ln with l(v) = fn f.v dn + fr 1 F.v df and r ~ 0. We extend this non-convex density energy, the classical strategy of searching for the saddle points of the augmented Lagrangian i.e. the (u, p, A.) solutions of the problem :

Lr(u, p, J.L) ~ Lr(u, p, A.)~ Lr(v, q, A.), Vv E Dad, Vq E H, VJ.L E H
To solve the initial problem, we use an algorithm inspired of by Uzawa method [START_REF] Letallec | Une methode numerique pour les problemes d'equilibre de corps hyperelastiques incompressibles en grandes deformations[END_REF] which involves a two-step iteration : a linear global problem and a non linear local problem.

THE "RELAXED" ENERGY

To profit by an existence theorem, we calculate the quasiconvexification QW of the previous potential by using the approach of Pipkin [START_REF] Pipkin | Elastic material with two preferred states[END_REF], and the convexification CW of W. To determine the quasiconvexification, we take a unit cube B, we use the definition of QW [START_REF] Pipkin | Elastic material with two preferred states[END_REF] QW(e) := lim i!lf

{ W(e + Y'suN) dx, UN-+0 umformly} B (2)
and consider a sequence of displacement fields UN for which the strain varies rapidly and periodically between two alternate constant values e1 and e2. The details of the calculus are given in the paper (Pagano et al., to appear).

UN is a periodic function of the variable n.x, of period !J.

Pipkin defines this sequence on one period (0, !J] by

UN := {2B(n.x)u on(O, 1 j\/] -2(1-B)(n.x-~)u on( 1 j\/',!J] { el=e+B(u0n+n0u) ande+Y'sUN= e2=e+(1-B)(u0n+n0u)
where u and n are two constant vectors and 0 < B < 1. We can prove that the density energy QW is :

QW(e) = min [(1-B)W(e 1 ) + BW(e 2 )]. (3) u,n,O~B9
This minimization can be performed by discrimination of all the possible cases for W(et) and W(e 2 ). We obtain a similar form, noted RW, for the convexification and the quasiconvexification of the density energy W. for The functional RW is equal to the convexification of W if 9 = 9c and to the quasiconvexification for g = 9q with :

9c = 2 min W_(E)-c] = -o: 2 7: K-1 7 EEM~x3 and 9q -2 min W_(u 0 n + n 0 u)-c] u,n
In particular, 9q = 9c if 7 has the form a 0 b + b 0 a where a and b are two vectors. If we restrict our attention to the case of isotropic Hooke's law with Lame coefficients .X and /1, and if we suppose that 7 = I, we can calculate the coefficients 9q and 9c to obtain the following results :

-0:2 -0:2

9c = and 9q = 2.X+!-l .X+ 2!-l
The optimal values of B which minimize (3), noted 0_ and B+ may be interpreted as two internal variables which represent the "proportions of martensite" of the mixture. These values are the following :

(1,0)

if 0:1 : e ~ -c + ~ ( 0:1 : e + c + ~ ) g ,0 if -c + 52 < 0:1 : e < -c -52 2 - - 2 (B-,B+) = (0,0) if o: 1 1 : e 1 ~ c + ~ ( 5) ( -0:1 : e + c + ~) 0, g if c + Q < 0:1 : e < c -52 2 - - 2 (0, 1) if 0:1 : e ~ c -~
For the case c + ~ ~ 0 we obtain, also, similar expressions for QW and CW (analogous to the Pipkin's result for a two-well energy). We see that QW and CW are piecewise quadratic and C 1 . From a well known result of Dacarogna, any solution of this relaxed problem is a limit of a minimizing sequence of the potential associated to the initial problem (Dacorogna, 1985). To solve the relaxed problem, we use the same method as for the initial problem.

Local minimum

The potential W, at fixed temperature, can appear as the difference of two convex functions (d.c.) (Pagano et al., to appear) 'Pl and y 2 :

'Pt(e) = Wo(e) and 'P2(e) = max(o:ltrel-c)

Based on this decomposition, we formulate the mechanical problem as two variational inequalities. In the finite dimensional case, this problem is equivalent to finding local minima of the potential energy. The decomposition leads to the introduction of a Lagrangian Ln, in the terminology of [START_REF] Auchmuty | Duality algorithms for nonconvex variational principles[END_REF]. It depends on the displacement field corresponding to a stress field, and it is function of the energy <I> 1 and the Fenchel convex conjugate function <f>2. This Lagrangian is separately convex in each of the dual variables, but the convexity of Ln is not guaranteed in general. An algorithm is derived to reach 8-critical points of the Lagrangians Ln. These points are defined as follows :

(u, cr 2 ) E VadxH is a 8-critical point of Ln if:

Ln(u, cr2) < Ln(v, cr2), Vv E Vad Ln(u, cr2) < Ln(u, 1), V1 E H

The relationship between critical points of the potential energy and those of Ln is somewhat more delicate. If ( u, cr2) E U adXH is a 8-critical point of the Lagrangian then u is a critical point of the potential energy.

This convex difference algorithm constructs minimizing sequence of Ln. It permits us to characterize a &-critical point of Ln. In practice, it determines local minima of the Lagrangian. It is also split into two steps, (global and local steps).

We remark that the "relaxed" energy can, also, be split into the difference of two convex functions cp 1 and Rcp2. The same numerical method can be applied to the resolution of the relaxed problem.

In this part, we compare the three points of view. We assume that r = I and we consider a flat sample with a geometrical defect submitted to a strain controlled loading-unloading process. For symmetry reasons, only a quarter of the sample is represented [Figure 3]. The incremental process is performed using loading steps. At each step the algorithm is initialised to the state at the end of the previous step.

With the initial energy W and t he D.C. approaches, we notice that a local non monotone strain-stress law (figure 1) leads to a monotone displacement-force diagram. Moreover a hysteresis behaviour is shown. For a given displacement value loading or unloading process, the phase transformation in the flat is not the same. The first picture in figure 4 represents the binary phases distribution in the flat. According to the chosen modelling, the proportions are either zero or one almost everywhere. Roughly speaking the phase transition is locally instantaneous.

With the global minimization , (quasiconvexified or not) , the hysteresis vanishes. The distribution of phases, issuing from the minimisation of the quasiconvexification, appears as the "homogenization" of the phase distribution obtained for the non convex potential W. Some differences are obtained about the spatial repartition of phases for the quasiconvexification (figure 4-2) and the convexification (figure 4-3). With CW, the zone where the volumic proportion of martensite lies between zero and one is more important than for relaxed energy QW. Aknowledgement : We thank Doctor Licht for his very interesting and helpful comments.

W 0

 0 (e, T) =~eKec(T)-C 0 TlogT W_(e, T) = W 0 (e, T) + a(T)r: e + c(T) W+(e, T) = Wo(e, T)-a(T)r: e + c(T) with a(T) = { 0 -a(T -Tc) ifthT ::; ~c and c(T) = !: (T -To) o erwzse <o

Figure 1 .

 1 Figure 1. Energy W for T :::; To, To :::; T :::; Tc and T ~ Tc

  diva+ f = 0 in n e(u) = Y'su innConstitutive law: a E DeW(e(u)) inn u = D on ro a.n = F on f1

  e ::;c + % Wo (e) + 2 ~ ( 0:7 : e + c + ~) 2 if -c + % ::; o:7 : e ::; -c -~ RW(e) = W 0 (e) if o:j7: ej ::; c + ~ (4) Wo (e) + 2 ~ ( 0:7 : e -c -%) 2 if c + ~ ::; 0:7 : e ::; c -~ W + (e) if 0:7 : e ~ c -~

-

  ar: e-c 2J ( CXT : e + C + ~) 2 (Rcp2) = 0 2J ( ar : e -c -~) 2 CXT: e-C if ar : e :::; -c + ~ if -c + Q < ar : e < -cair : el :S c + ~ if C + ~ :::; CXT : e :::; -~ if CXT : e ;?: C -~ 6. Numerical simulations at fixed temperature

Figure 2 .

 2 Figure 2. Flat sample


Hélène Sportouche

Antoine Roueff

Pascale C Dubois-Fernandez

Precision of Vegetation Height Estimation using the Dual-Baseline PolInSAR System and RVoG Model with Temporal Decorrelation

Keywords: Polarimetric SAR Interferometry (PolInSAR), dualbaseline, Random Volume over Ground (RVoG) model, temporal decorrelation, vegetation height estimation, Cramer-Rao bound (CRB)

Estimating vegetation height from PolInSAR data using the RVoG model has motivated several studies. Most of these propose estimators and apply them to real data to demonstrate their potential. In previous publications on the single-baseline system, we proposed a complementary approach which consisted in analyzing the precision of estimations of vegetation height that can be expected depending on the considered model and on the available a priori knowledge. In this paper, we develop such an analysis for the case of a dual-baseline system.

 50] m, and for polarimetric contrast between the ground and the volume larger than 0.3. Furthermore, we investigate the performance of a Maximum-Likelihood estimator and compare this to the precision given by the CRB. For the examples considered, with N = 200 pixels, we observed convergence issues of the estimator when the polarimetric contrast is smaller or equal to 0.3.

I. INTRODUCTION

Worldwide biomass monitoring is essential for a better understanding of the carbon cycle and can be performed from synthetic aperture radar (SAR) measurements [START_REF] Le Toan | The biomass mission: Mapping global forest biomass to better understand the terrestrial carbon cycle[END_REF]. Because the radar backscattered signal saturates for high biomass depending on the radar frequency, a direct biomass estimation from intensity data is not always possible. An alternative approach consists in first estimating forest height using polarimetric interferometric SAR (PolInSAR) techniques [START_REF] Cloude | Polarimetric SAR interferometry[END_REF] and then estimating the biomass through allometric relationships [START_REF] Le Toan | The biomass mission: Mapping global forest biomass to better understand the terrestrial carbon cycle[END_REF]. Being able to correctly retrieve the vegetation height is thus of great importance for biomass estimation.

Different methods have been proposed in the literature to estimate vegetation height from PolInSAR data, depending on the considered configuration and on the available a priori knowledge. In the single-baseline (SB) PolInSAR configuration, corresponding to the acquisition of two PolSAR measurements, vegetation height estimators have been successfully applied on real data at different bands such as L and P [START_REF] Papathanassiou | Single-baseline polarimetric SAR interferometry[END_REF], [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF], [START_REF] Papathanassiou | The effect of temporal decorrelation on the inversion of forest parameters from Pol-InSAR data[END_REF], [START_REF] Garestier | Forest Height Inversion Using High-Resolution P-Band Pol-InSAR Data[END_REF], [START_REF] Lavalle | Extraction of structural and dynamic properties of forests from polarimetric-interferometric SAR data affected by temporal decorrelation[END_REF]. In most cases, these estimators are based on the Random Volume Over Ground (RVoG) model, introduced by Treuhaft et al [START_REF] Treuhaft | Vertical structure of vegetated land surfaces from interferometric and polarimetric radar[END_REF]. To build an estimator, Tabb et al proposed to use a maximumlikelihood based approach [START_REF] Flynn | Direct estimation of vegetation parameters from Covariance data in polarimetric SAR interferometry[END_REF], [START_REF] Tabb | Full maximum likelihood inversion of PolInSAR scattering models[END_REF], which allows one to estimate not only vegetation and ground heights but also polarimetric coherency matrices of the ground and of the volume. Another estimation approach proposed in [START_REF] Papathanassiou | Single-baseline polarimetric SAR interferometry[END_REF] by Papathanassiou et al provides in addition a geometric interpretation of the complex coherence observed for different polarization states. Afterwards, several variants of the estimator from [START_REF] Papathanassiou | Single-baseline polarimetric SAR interferometry[END_REF] have been developed, depending on the considered model and on the available a priori knowledge. Initially in [START_REF] Papathanassiou | Single-baseline polarimetric SAR interferometry[END_REF], [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF], the authors proposed to estimate vegetation height and extinction coefficient, under the assumption that, at L-band, the effective ground-tovolume ratio is equal to zero for one polarimetric channel in order to solve the ambiguity problem encountered for this kind of SB approaches with such a model [START_REF] Hajnsek | Tropical-Forest-Parameter Estimation by Means of Pol-InSAR: The INDREX-II Campaign[END_REF], [START_REF] Lee | The impact of temporal decorrelation over forest terrain in polarimetric SAR interferometry[END_REF]. Alternatively, at P-band, when the ground has a significant contribution for all polarizations, SB height estimation is performed in [START_REF] Garestier | Forest Height Inversion Using High-Resolution P-Band Pol-InSAR Data[END_REF], [START_REF] Kugler | Frequency effects in Pol-InSAR forest height estimation[END_REF], assuming that the extinction coefficient is known a priori.

Furthermore, it has been shown in [START_REF] Lee | The impact of temporal decorrelation over forest terrain in polarimetric SAR interferometry[END_REF], [START_REF] Dubois-Fernandez | The TropiSAR airborne campaign in French Guiana: Objectives, Description and Observed Temporal behavior of the Backscatter signal[END_REF] that the temporal decorrelation phenomenon, which often occurs in the repeat-pass scenario, can have a significant influence on the height retrieval and thus should be taken into account. In [START_REF] Papathanassiou | The effect of temporal decorrelation on the inversion of forest parameters from Pol-InSAR data[END_REF], the RVoG model with volume temporal decorrelation (called RVoG+VTD) has been introduced and allows the estimation of the vegetation height and of a temporal decorrelation coefficient in SB, assuming not only a negligible ground-to-volume ratio, but also an a priori known extinction coefficient [START_REF] Krieger | Spaceborne polarimetric SAR interferometry: performance analysis and mission concepts[END_REF]. More recently, another alternative to address temporal decorrelation has been proposed based on a Random Motion over Ground (RMoG) model [START_REF] Lavalle | A temporal decorrelation model for polarimetric radar interferometers[END_REF], [START_REF] Lavalle | Extraction of structural and dynamic properties of forests from polarimetric-interferometric SAR data affected by temporal decorrelation[END_REF]. In these articles, Lavalle et al, generalizing the works of Zebker and Villasenor [START_REF] Zebker | Decorrelation in interferometric radar echoes[END_REF], consider unstable scatterers located at the ground layer as well as in the volume layer, with a motion variance changing along the vertical axis. In [START_REF] Lavalle | Extraction of structural and dynamic properties of forests from polarimetric-interferometric SAR data affected by temporal decorrelation[END_REF], the authors use an objective function with five observations of the temporal and volumetric complex coherence, to inverse in SB the vegetation height, the extinction coefficient and the motion standard deviations of ground and canopy elements from real airborne SAR data at L-band. In [START_REF] Lei | Estimation of forest height using spaceborne repeat-pass L-band InSAR correlation magnitude over the US State of Maine[END_REF], Lei et al combine the RMoG model with the effects of dielectric changes for large temporal baselines and provide a simplified height inversion procedure from SB HV-polarized InSAR correlation magnitude data, giving an alternative approach when full-polarization data are not available.

To complement these single-baseline approaches, dualbaseline (DB) configurations have also been considered [START_REF] Cloude | Robust parameter estimation using dual baseline polarimetric SAR interferometry[END_REF], [START_REF] Cloude | A coherent EM scattering model for dual baseline POLInSAR[END_REF], [START_REF] Shunjun | Robust vegetation height extraction using maximum likelihood estimation for dual-baseline PolInSAR[END_REF], [START_REF] Zhou | Analysis of temporal decorrelation in dual-baseline POLinSAR vegetation parameter estimation[END_REF], [START_REF] Lee | Quantification and compensation of temporal decorrelation effects in polarimetric SAR interferometry[END_REF], [START_REF] Lavalle | Three-baseline InSAR estimation of forest height[END_REF]. In [START_REF] Cloude | Robust parameter estimation using dual baseline polarimetric SAR interferometry[END_REF] and [START_REF] Cloude | A coherent EM scattering model for dual baseline POLInSAR[END_REF], the introduction of a second spatial baseline allows the development of two DB methods under the RVoG model (respectively a geometrical one and a formal one with norm minimization) using at least four complex coherence observations (i.e. considering at least two polarization channels per baseline) for vegetation height and extinction coefficient estimation, without the zero ground-to-volume ratio hypothesis applied in SB [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF]. Moreover, in [START_REF] Zhou | Analysis of temporal decorrelation in dual-baseline POLinSAR vegetation parameter estimation[END_REF] and in [START_REF] Lee | Quantification and compensation of temporal decorrelation effects in polarimetric SAR interferometry[END_REF], two DB inversion approaches (respectively a geometrical one and a formal one with vectorial criterion minimization), developed under the RVoG+VTD model and also based on complex coherence observations, enable the simultaneous estimation of vegetation height, extinction coefficient and two volume temporal decorrelation factors, but this time conserving a zero ground-to-volume ratio hypothesis. In [START_REF] Lavalle | Three-baseline InSAR estimation of forest height[END_REF], a DB inversion approach, developed under the RMoG model and using six complex coherence observations provided by the three available spatial baselines, enables to simultaneously estimate vegetation height, extinction coefficient and six temporal decorrelation parameters.

Finally, multi-baseline (MB) approaches with larger number of acquisitions have also been considered. In [START_REF] Neumann | Estimation of Forest Structure, Ground, and Canopy Layer Characteristics From Multibaseline Polarimetric Interferometric SAR Data[END_REF], several forest characteristics (including in particular a temporal decorrelation parameter for each baseline) are retrieved from MB PolInSAR data by a joint exploitation of polarimetric and interferometric diversities. In [START_REF] Lee | Quantification of temporal decorrelation effects at L-band for polarimetric SAR interferometry applications[END_REF], two temporal decorrelation coefficients relative to the ground and volume layers are quantified and their impacts on MB PolInSAR vegetation height estimation are studied. Moreover, in [START_REF] Treuhaft | Vegetation profiles in tropical forests from multibaseline interferometric synthetic aperture radar, field, and lidar measurements[END_REF] and [START_REF] Treuhaft | The ambiguity in forest profiles and extinction estimated from multibaseline interferometric SAR[END_REF], MB InSAR coherence and phase observations are used for the retrieval of vertical vegetation density profiles. Eventually, MB InSAR or PolInSAR data have been also widely analyzed to develop tomographic techniques [START_REF] Reigber | First demonstration of airborne SAR tomography using multibaseline L-band data[END_REF], [START_REF] Lombardini | Adaptive spectral estimation for multibaseline SAR tomography with airborne L-band data[END_REF], [START_REF] Tebaldini | Algebraic synthesis of forest scenarios from multibaseline PolInSAR data[END_REF], [START_REF] Tebaldini | Single and Multipolarimetric SAR Tomography of Forested Areas: A Parametric Approach[END_REF], [START_REF] Hajj Chehade | Polarimetric tomography for structural characterization of tropical forest[END_REF], [START_REF] Ferro-Famil | ML tomography based on the MB RVoG model: Optimal estimation of a covariance matrix as a sum of two Kronecker products[END_REF], requiring usually at least six PolSAR measurements.

Most of these analysis focus on proposing estimation algorithms which are applied to real data to demonstrate their potential, using in situ measurements or LIDAR measurements as ground truth. In this paper, as in the recent studies in [START_REF] Roueff | Cramer-Rao lower bound analysis of vegetation height estimation with random volume over ground model and polarimetric SAR interferometry[END_REF], [START_REF] Arnaubec | Influence of the nature of a priori knowledge on the precision of vegetation height estimation in polarimetric SAR interferometry[END_REF], [START_REF] Réfrégier | Invariant Contrast Parameters of PolInSAR and Homogenous RVoG Model[END_REF], we propose a complementary approach which consists in analyzing the precision of vegetation height estimation depending on the considered model and on the available a priori knowledge. More precisely, we explore the amount of information present in the data based on the Cramer-Rao bound (CRB) as a function of the system configuration and as a function of some vegetation characteristics. In particular, we analyse the sensitivity of the CRB with respect to the vertical wavenumbers, the vegetation height, the extinction coefficient, the volume coherence and the polarization properties of the volume and of the ground. The CRB is a bound on the variance of unbiased estimators [START_REF] Garthwaite | Statistical Inference[END_REF] for a given model and a priori knowledge. It is often used as a reference for precision. It does not depend on the choice of the considered estimator. In [START_REF] Roueff | Cramer-Rao lower bound analysis of vegetation height estimation with random volume over ground model and polarimetric SAR interferometry[END_REF], it has been shown that the CRB of the vegetation height can be very high in SB, and thus that the corresponding precision on the vegetation height can be very poor, but in [START_REF] Arnaubec | Influence of the nature of a priori knowledge on the precision of vegetation height estimation in polarimetric SAR interferometry[END_REF] it has also been shown that some a priori knowledge can allow the precision to be improved significantly. More recently in [START_REF] Réfrégier | Invariant Contrast Parameters of PolInSAR and Homogenous RVoG Model[END_REF], it has been shown that the CRB of the vegetation height depends only on four unknown parameters, which allows one to identify the contrast parameters which influence the precision of the considered estimation problem.

In this paper, we apply this methodology based on the CRB to the dual-baseline configuration to explore whether adding a third PolSAR image allows one to estimate accurately the vegetation height when extinction coefficient and temporal coherence are unknown. Furthermore, we investigate whether a maximum-likelihood (ML) estimator can reach the precision given by the CRB. Thus, we consider a DB PolInSAR model, which takes into account temporal decorrelation of the volume, based on the RVoG+VTD model [START_REF] Papathanassiou | The effect of temporal decorrelation on the inversion of forest parameters from Pol-InSAR data[END_REF], with unconstrained polarimetric coherency matrices.

We show via simulations that, the precision described by the vegetation height CRB is sensitive not only to system configuration (vertical wavenumbers), but also to the vegetation characteristics (in particular vegetation height and polarimetric properties). Thus, optimizing the system to obtain a one meter precision on vegetation height estimation for all vegetations seems impossible. However, we show an operating regime exists for which the vegetation height estimation precision is around 1 m for N = 200 pixels and for vegetation heights between 20 and 50 m, which corresponds to a large percentage of the higher biomass forest. Section II details the model, the CRB computation and the implemented maximum likelihood (ML) estimator. Then, the evolutions of the vegetation height CRB as a function of the system configuration and of the vegetation characteristics are analyzed in section III. Finally, in section IV, the main conclusions of the paper are drawn and some perspectives are considered.

II. CRAMER-RAO BOUND AND MAXIMUM-LIKELIHOOD BASED ESTIMATOR WITH THE DUAL-BASELINE CONFIGURATION AND RVOG MODEL

A. Dual-baseline RVoG model

For each pixel of a DB PolInSAR image, a 9D complex valued vector k is defined as the concatenation of three PolSAR measurements

u i by k = [u T 1 , u T 2 , u T 3 ]
T , where u i can be written in the Pauli basis as

u i = 1 √ 2 [S HH i + S V V i , S HH i -S V V i , 2S HV i ]
T and where S Y X i denotes the complex scattering amplitude measured at antenna i for a given X emitting polarization state and a given Y receiving polarization state. Introducing T ij = u i u † j , where . is the statistical expectation operator, the 9 × 9 covariance matrix of k has the following form:

Υ = k (k) † =   T 11 T 12 T 13 T † 12 T 22 T 23 T † 13 T † 23 T 33   . ( 1
)
where

(T ij ) (i,j)∈{1,2,3} 2 are 3 × 3 matrices.
We assume that the vegetation can be described by a homogeneous Random Volume over Ground (RVoG) model [START_REF] Treuhaft | Vertical structure of vegetated land surfaces from interferometric and polarimetric radar[END_REF] with temporal decorrelation in the volume layer similarly to [START_REF] Papathanassiou | The effect of temporal decorrelation on the inversion of forest parameters from Pol-InSAR data[END_REF]. Thus, introducing ρ ij as the temporal correlation coefficient (equivalently (1ρ ij ) is called the temporal decorrelation coefficient [START_REF] Papathanassiou | The effect of temporal decorrelation on the inversion of forest parameters from Pol-InSAR data[END_REF]) between antenna i and antenna j (where ρ ii = 1 ∀i ∈ {1, 2, 3}), k z ij as the vertical sensitivity (or vertical wavenumber) between antenna i and antenna j (where k z ii = 0 ∀i ∈ {1, 2, 3}), and T vol and T gro as respectively the polarimetric coherency matrices associated with volume and ground, for (i, j) ∈ {1, 2, 3} 2 , the matrix T ij can be written:

T ij = e jk z ij zij (ρ ij I ij T vol + aT gro ) . (2)
Note that with this model, the coherency matrices T vol and T gro do not depend on the baseline ij. Moreover, in Eq. ( 2), I ij and a are defined by:

I ij = e jk z ij hv -e -αhv jk z ij +α , a = e -αhv with α = 2σ v / cos β, (3)
where σ v is the polarization-independent and vertically constant extinction coefficient of the volume, h v is the vegetation height and β is the incident angle of the radar [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF]. Note that in Eq. ( 2), the ground height parameter z ij depends on the indices (i, j). Obviously, the true ground height does not depend on the interferometric indices (i, j). However, due to phase calibration issues, the interferometric phase associated with the indices (i, j) may introduce an error in the measured ground height [START_REF] Dinh | Relating P-band synthetic aperture radar tomography to tropical forest biomass[END_REF], [START_REF] Tebaldini | Phase Calibration of Airborne Tomographic SAR via Phase Center Double Localization[END_REF]. As a result, knowledge on the SB ground phase cannot be used as input information for another pair as it can be affected by this phase bias. To take into account this uncertainty, we introduce a ground height z ij that depends on the interferometric pair.

In the considered model, the system parameters β and k z ij {(i,j)=(1,2),(2,3),(1,3)} are known from the experimental setup, while the other parameters are unknown and have to be estimated in an homogeneous region where measurements are identically distributed. Moreover, introducing the interferometric ground phase

φ ij = k z ij z ij , it can be shown [41] that k z 13 = k z 12 + k z 23 and φ 13 = φ 12 + φ 23 , thus z 13 = 1 k z 13 (k z 12 z 12 + k z 23 z 23 ) , (4)
which means that the number of unknown ground heights (or similarly ground phases) can be reduced to two.

In this paper, we generate data for which numerical values of parameters verify the particular case:

z 12 = z 23 and ρ ij = ρ for i = j. (5)
We make this assumption to be able to compare the estimation precisions as a function of different a priori knowledges: situations for which estimations are obtained using the knowledge of both equations in Eq. ( 5), but also when it is not the case. In particular, when the estimator does not use this a priori knowledge, it may estimate three temporal coherences and two ground heights. The four considered situations correspond to different vectors of unknown parameters:

θ 22 = {(t vol,k ) k=1,..,
where the index n on θ n indicates the number of unknown parameters and (t m,k ) k=1,..,9 denote the 9 unknown real coefficients which parameterize the matrix T m for m ∈ {vol, gro}. Note that it is possible to use symmetry properties to constrain the matrices T vol and T gro , and thus decrease the number of unknown parameters. However, such a priori knowledge is not useful in single-baseline [START_REF] Arnaubec | Influence of the nature of a priori knowledge on the precision of vegetation height estimation in polarimetric SAR interferometry[END_REF] to improve the vegetation height estimation, and seldom used in practice [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF]. Thus, we choose not to use any assumption on T vol and T gro .

B. CRB computation

It has been shown in [START_REF] Roueff | Cramer-Rao lower bound analysis of vegetation height estimation with random volume over ground model and polarimetric SAR interferometry[END_REF] that the CRB can be useful to characterize the estimation precision in the SB PolInSAR configuration. This paragraph summarizes the CRB properties previously presented in [START_REF] Roueff | Cramer-Rao lower bound analysis of vegetation height estimation with random volume over ground model and polarimetric SAR interferometry[END_REF]. Consider a set χ = {k 1 , k 2 , ..., k N } of N measurements of a DB PolInSAR image. Assume that the vector of unknown parameters is θ = {(θ p ) p=1,...,K } (it can be θ 22 , θ 23 , θ 24 or θ 25 but we removed the index to simplify notations) and consider an estimator θ p (χ) of the parameter θ p . The mean and variance of θ p (χ) are respectively denoted by µ θp and σ 2 θp . The precision of an unbiased estimator (i.e. for which µ θp = θ p ) can be described by its variance σ 2 θp . However, before looking for an estimator θ p (χ), it can be interesting to analyze CRB N [θ p ] (i.e. the CRB of θ p for a sample of size N ), because it provides a lower bound on the variance σ 2 θp of unbiased estimators (i.e. σ 2 θp ≥ CRB N [θ p ]). As explained in [START_REF] Garthwaite | Statistical Inference[END_REF], CRB N [θ p ] is defined as the element (p, p) of the inverse of the Fisher information matrix I N (θ), which is defined by [I N (θ)] (p,q)∈{1, ..., 23} 2 = -(∂ 2 /∂θ p ∂θ q ) log P (χ|θ) , where P (χ|θ) is the probability density function of observing χ when the parameter is θ. In this paper, measurements k n are assumed to be statistically independent and identically distributed with a zero mean complex circular Gaussian with covariance matrix Υ, which means that :

P (χ|θ) = 1 π 9N det(Υ) N exp(-N tr(Υ -1 Υ)) (7)
where tr is the trace, det is the determinant, Υ =

1 N N n=1 k n k n
† is the empirical covariance matrix obtained from the measurements and Υ is the covariance matrix parameterized by the vector of unknown parameters θ. In which case, the element (p, q) of the matrix I N (θ) can be written [START_REF] Stoica | Spectral Analysis of Signals[END_REF] as

[I N (θ)] p,q = N tr Υ -1 ∂Υ ∂θ p Υ -1 ∂Υ ∂θ q , (8)
The expressions of the partial derivatives ∂Υ ∂θp required to calculate CRB N [θ p ] are straightforward generalizations of those described in [START_REF] Roueff | Cramer-Rao lower bound analysis of vegetation height estimation with random volume over ground model and polarimetric SAR interferometry[END_REF] and thus are not detailed in this paper.

Since the CRB only gives a bound of the variance of unbiased estimator, it is also important to check whether the bound is reached by an estimator. For this purpose, an iterative maximum-likelihood based estimator is proposed. The implementation of this estimator is detailed in the next section.

C. Implemented maximum-likelihood estimator

When the precision of vegetation height estimation in single-baseline was analyzed in [START_REF] Roueff | Cramer-Rao lower bound analysis of vegetation height estimation with random volume over ground model and polarimetric SAR interferometry[END_REF], the Cloude et al's estimator proposed in [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF] was observed to be an efficient estimator, which means that it is unbiased and its variance is equal to the CRB, when the sample size N is sufficiently high. In this paper, a maximum-likelihood estimator is used in dual-baseline to provide a simple criterion taking into account all the available baselines. The maximumlikelihood estimator, previously used in [START_REF] Tabb | Full maximum likelihood inversion of PolInSAR scattering models[END_REF] in singlebaseline PolInSAR configuration, is a standard algorithm which can be used in most of estimation problems (see [START_REF] Garthwaite | Statistical Inference[END_REF] for a didactic tutorial).

In the PolInSAR literature [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF], [START_REF] Hajnsek | Tropical-Forest-Parameter Estimation by Means of Pol-InSAR: The INDREX-II Campaign[END_REF], [START_REF] Lee | Quantification of temporal decorrelation effects at L-band for polarimetric SAR interferometry applications[END_REF], [START_REF] Lavalle | Three-baseline InSAR estimation of forest height[END_REF], [START_REF] Lavalle | Extraction of structural and dynamic properties of forests from polarimetric-interferometric SAR data affected by temporal decorrelation[END_REF], most of the estimators are based on the parametrization of coherence γ ij (w) =

w † T ij w √ w † T iiw √ w † T jj w
for different pairs of polarizations described by the vector w. The maximum-likelihood estimator is based on the minimization of the opposite of the log-likelihood:

J(θ) = N ln (det(Υ)) + N tr Υ -1 Υ , (9)
where the covariance matrix Υ is parameterized by the vector θ as previously described. The minimization of the function J(θ) allows one to estimate the unknown vector θ. In this paper, the optimization is performed with the Fisher's scoring algorithm [START_REF] Garthwaite | Statistical Inference[END_REF], which is based on the recursive scheme:

θ n+1 = θ n + p n d n with d n = -(I N ( θ n )) -1 ∇ θ J( θ n ), (10)
where, at iteration n, the vector θ n is an estimation of the vector θ, the scalar p n is the step (herein chosen equal to 0.1), and the vector ∇ θ J denotes the gradient of J(θ). The p th component of

∇ θ J is equal to ∂J(θ) ∂θp = N tr Υ -1 ∂Υ ∂θp -tr Υ -1 ∂Υ ∂θp Υ -1 Υ [42].
In such an iterative procedure, it is often crucial to initialize it with a vector θ 0 not too far from the global minima of J(θ) because of the presence of local minima. A possible initialization procedure (i.e. estimation of θ 0 ), which relies on single-baseline approaches, is presented in appendix A and in appendix B when one estimates only one coherence ρ (i.e. one uses the a priori knowledge that ρ ij = ρ for i = j). It is shown in the next section III-A that the CRB of the vegetation height are so large with three unknown coherences that it appeared unnecessary to implement an estimator in this case. Furthermore, with the scoring algorithm, it can also be important to regularize the inversion of the matrix I N ( θ n ) in Eq. ( 10) with a pseudoinverse [START_REF] Horn | Topics in matrix analysis[END_REF] (in our implementation, the limit is when the ratio between the largest and the smallest singular value is higher than 10 16 ).

Another issue is to decide when the optimization should stop. With the proposed implementation, there are 3 situations which stop the recursive scheme. The iteration stops when the quantity 1 N |J( θ n ) -J( θ n-1 )| is inferior to a given value (chosen equal to 10 -6 in this paper), or when the number of iterations n max = 1000 is reached, or when the optimization algorithm is providing parameter values leading to a non-defined criterion value J( θ n ). This last situation usually occurs because the estimated covariance matrix T vol or T gro has negative eigenvalues. In this case, the estimator is considered to fail. In addition, if the obtained value h v is with an error | h vh v | larger than 6 m, then the estimator is also considered to fail. This arbitrary value of 6 m allows us to define outliers (i.e. bad estimates), whose proportion is analyzed in the simulation section with the so-called estimated "success rate".

D. Simulated RVoG model parameters

A straightforward adaptation of the result obtained in [START_REF] Réfrégier | Invariant Contrast Parameters of PolInSAR and Homogenous RVoG Model[END_REF] for the single-baseline case, to the dual-baseline configuration shows that the CRB of the vegetation height and the variance of the maximum likelihood estimator depends on the form of T vol and T gro only through the 3 eigenvalues of T -1 vol T gro denoted λ n (with λ 1 > λ 2 > λ 3 ). In other words all datasets for which T -1 vol T gro are equal to (λ n ) n=1,2,3 (with identical values for h v , σ v , ρ, ...) have the same vegetation height estimation precision. Thus, in the simulations of this paper, we generated synthetic data with

T vol =   1 0 0 0 1 0 0 0 1   T gro =   λ 1 0 0 0 λ 2 0 0 0 λ 3   (11
) Furthermore, the adaptation of [START_REF] Réfrégier | Invariant Contrast Parameters of PolInSAR and Homogenous RVoG Model[END_REF] in dual-baseline also shows that the vegetation height estimation precision does not depend on the values of z 12 and z 23 (even in the situation with z 12 = z 23 ). Thus, in the simulations of this paper, we generated synthetic data with

z 12 = z 23 = 1 m ( 12
)
The parameters that influence the precision of vegetation height estimation are β, k z 12 , k z 23 , h v , σ v , ρ and (λ n ) n=1,2,3 . Exploring the whole space of dimension 9 is not possible. In section III, first, the variation of the CRB of h v as a function of k z 12 , k z 23 is analyzed, to discuss the potential optimization of the system. Next, the variation of the CRB of h v is analyzed as a function of

h v , σ v , ρ, A = (λ 1 -λ 3 )/(λ 1 + λ 3 ) and E = λ 1 + λ 2 + λ 3 .
As explained in [START_REF] Réfrégier | Invariant Contrast Parameters of PolInSAR and Homogenous RVoG Model[END_REF], the parameter A is related to the difference between the polarimetric properties of T vol and those of T gro . On the one hand, A = 0 implies that T vol and T gro are proportional and coherence γ ij (w) are independent of w. On the other hand, A 1 implies that a pair of polarization exists, described by a vector w, for which the ground response contribution is null (i.e. for which

γ ij (w) = e jk z ij zij ρij Iij √ IiiIjj ).
The parameter E is related to the ground-to-volume power ratio between the wave backscattered by the ground and the one backscattered by the volume.

The relation between (λ n ) n=1,2,3 and A, E, X is given by

λ 1 = E(1 + A)/(3 -A + 2AX ) λ 2 = E(1 -A + 2AX )/(3 -A + 2AX ) λ 3 = E(1 -A)/(3 -A + 2AX ) (13)
where X = (λ 2λ 3 )/(λ 1λ 3 ).

III. PRECISION ANALYSIS

A. Evolution of the CRB as a function of the system configuration

Figure 1 shows the evolution of the CRB of h v as a function of the system configuration for different a priori knowledge (see Eq. ( 6)) and for N = 200 pixels. For all considered a priori knowledge, when k z 12 = 2πn/h v and k z 23 = 2πm/h v with (n, m) a pair of strictly positive integers, the precision described by CRB 1/2 N (h v ) becomes larger than 10 m (these particular values are denoted by white dashed lines in Figure 1(a-d)). Thus, for these values, it is impossible to have an accurate estimation of h v . In the rest of the space (k z 12 , k z 23 ), the value of CRB ). Thus, it seems impossible to estimate the vegetation height accurately when 3 temporal coherences are unknown. For the rest of the paper, we thus focus on situations (c) and (d), for which only one temporal coherence ρ is estimated. Moreover, based on Figure 1, we choose k z 12 = 0.06 m -1 and k z 23 = 0.25 m -1 so that the CRB of h v remains small for h v ∈ [10, 50] m.

Furthermore, figure 1 (c) and (d) show that knowing a priori that z 12 = z 23 can also allow the reduction of the CRB of h v compared to the case where this a priori is unknown. In the following, we continue to compare these two cases. In the next paragraph, the proposed maximum likelihood estimator is thus analyzed in both scenarios where θ 22 or θ 23 are estimated.

B. Evolution of the CRB and of the proposed estimator performances as a function of h v and A

This section presents an analysis of the performances of the estimators obtained with Monte-Carlo simulations. For each example considered, P = 200 realisations of the sample χ (p) = {k 1) and ( 2)) assuming a Gaussian circular distribution. Then, the empirical root mean square errors (RMSE) rmse

2 hv = 1 P p h v (χ (p) ) -h v 2
are computed. One also could have analyzed bias and variance separately, but we preferred to add these two sources of error and concentrate on the presence of outliers with the analysis of the percentage of success of the estimator. The size of sliding windows in PolInSAR images is usually around 200 pixels, we choose to analyze the performances for N = 200 and N = 2000 pixels. Furthermore, the iterative estimator performance are analyzed for two cases:

• When the initial estimation θ 0 is obtained with the algorithm presented in Appendix A. • When the initial estimation θ 0 is fixed to the true vector θ. Obviously this second initialization cannot be used with real data. The purpose is to analyze the performance of the estimator when the iterative estimator is initialized near the solution and thus is less sensitive to local maxima.

The results of the Monte Carlo simulation presented in Figure 2 (a) and (b) show that the percentage of success is always close to 100% when the initialization is at the solution, but it is not the case with the initialization proposed in Appendix A. When A = 0.3 (i.e. small polarization contrast) and h v = 28 m, these pourcentages are around 60% for both estimator (a) and (b). !"# N (h v ) varies a lot with h v , and in particular that CRB 1/2 N (h v ) increases when the vegetation height is smaller than 15 m. Thus, the chosen pair of vertical wavenumber is not adequate for small vegetations. Moreover, a comparison between Figure 2 (c) and(d) shows that CRB 1/2 N (h v ) can be bigger when two ground heights are unknown instead of one. For A = 0. 3 andh

2π/h v 4π/h v $"# %"# &"# 0 2 4 6 8 10 %'(')$!)# +∞ k z 12 k z 23 2π/h v 4π/h v 2π/h v 4π/h v k z 12 k z 23 2π/h v 4π/h v 2π/h v 4π/h v k z 12 k z 23 2π/h v 4π/h v 2π/h v 4π/h v k z 12 k z 23 2π/h v 4π/h v
v = 30 m, the CRB 1/2 N (h v ) increases from 0.7 m (in d) to 2 m (in c).
It seems that the influence of the polarimetric contrast A on CRB 1/2 (h v ) remains modest when only one ground height is unknown, whereas it is not the case when two ground heights are unknown.

Figure 2 (c) and (d) also show that the root mean square error (RMSE) computed with estimations with an error smaller than 6 m are close to the precision given by CRB 1/2 (h v ), especially for large values of A. Thus, the main issue of the estimator is the amount of outliers described by the pourcentage of success. These results are confirmed by Figure 3, where simular observations are obtained for A = 0.1 and A = 0.6.

The problem that occur for A = 0.1 with the initialization proposed in Appendix A (see the pourcentage of success smaller than 60% in Figure 3(a) and(b)) probably comes from the inaccuracy of single-baseline estimations [START_REF] Réfrégier | Invariant Contrast Parameters of PolInSAR and Homogenous RVoG Model[END_REF]. In such situations, one alternative is to increase the number of pixels. Figure 4 shows that when N = 2000, the pourcentage of success increases compared to the situation with N = 200. However, this supposes one can consider large homogeneous region of vegetation in the considered image.

To summarize, the precision given by the CRB is encouraging, but the proposed estimators may not be efficient when A is too small. Finding some optimization algorithms to obtain a better estimator is a motivating perspective. Another solution is to increase the window

h v [m] h v [m] A = 0.3 A = 0.9 A = 0.9 A = 0.3 A = 0.3 A = 0.9 h v [m] h v [m] % A = 0.3 A = 0.9 % CRB 1/2 N (h v ), rmse( h v ) [m] CRB 1/2 N (h v ), rmse( h v ) [m] (a) (b) (c) (d)
Fig. 2: Performances of the vegetation height estimators, as a function of h and for different values of A (A = 0.3 and A = 0.9). In (a) and (c), estimations have been obtained when 2 ground heights are estimated (i.e. θ 23 in Eq. ( 6)), whereas in (b) and (d), estimations have been obtained when only 1 ground height is estimated (i.e. θ 22 in Eq. ( 6)). In N (h v ), the sign + stands for estimations obtained with the scoring algorithm when the initial estimations θ 0 are given by the algorithm described in appendix A, and circle o stands for the one obtained when θ 0 is fixed to the true value. Large + and o are obtained for A = 0.9 and small ones for A = 0.

h v [m] h v [m] h v [m] h v [m] % % CRB 1/2 N (h v ), rmse( h v ) [m] CRB 1/2 N (h v ), rmse( h v ) [m] (a) (b) (c) (d)
h v [m] h v [m] h v [m] h v [m] % % CRB 1/2 N (h v ), rmse( h v ) [m] (a) (b) (c) (d) A = 0.1 A = 0.6 A = 0.1 A = 0.6 A = 0.6 A = 0.1 CRB 1/2 N (h v ), rmse( h v ) [m] A = 0.6 A = 0.1
Fig. 4: Same as Figure 3, but for N = 2000 pixels.

size, however this solution can lead to inhomogeneous samples. Thus, the analysis of the estimator performance in presence of inhomogeneities should be explored to analyze the bias induced by these inhomogeneities. These two aspects are not in the scope of this paper but open challenging perspectives. In the following, we choose A = 0.3 and investigate the influence of other vegetation parameters on CRB 1/2 (h v ).

C. Influence of other vegetation parameters

In this paper, we propose a methodology to analyze the precision that can be expected on the vegetation height estimation with dual-baseline configuration and with RVoG model without knowing a priori the extinction coefficient σ v and in presence of an unknown temporal decorrelation of the volume (described herein by a single coherence parameter ρ). The previously presented results show that the precision described by the square root of the CRB varies significantly as a function of h v . Investigating the physical reasons of these variations is a major issue that require further investigations. In the following, we analyse the influence of different parameters on the CRB: the parameter E, the attenuation σ v and the temporal coherence parameter ρ.

Figure 5 shows that when two ground heights are unknown, some local maxima appear at several values of h v . The intensity and the location of these local maxima change with E and σ v . When data are calibrated such that z 12 = z 23 , the importance of these local maxima is reduced significantly. This observation is in favor of calibrating the data phases before estimating h v .

Furthermore, an increase from E = 50 m -1 to E = 200 m -1 (i.e. an increase of the ground-to-volume power ratio of a factor 4) make the CRB of h v decrease for h v > 30 m, but an increase from E = 200 m -1 to E = 2000 m -1 seems to induce some CRB increases, especially with two unknown ground heights. This can be interpreted as a tradeoff on the ground-to-volume energy ratio. For the extinction coefficient σ v , it seems that the lowest CRB occurs when σ v ∈ [0.023, 0.046]. When σ v = 0.069m -1 , the vegetation height CRB increase rapidly after 50 m, probably because the ground is not visible anymore. Note that these phenomena also depend on A and on the other parameters. For temporal coherence ρ, the situation looks simpler. For h v ≤ 60 m, the CRB seem to be approximately a function of 1/ρ.

IV. CONCLUSION AND FUTURE WORK

We proposed a methodology to analyze the precision on the vegetation height estimation using the PolInSAR system in the DB configuration with an RVoG model, where the temporal decorrelation on the volume is unknown and where there is no a priori knowledge on the extinction coefficient. The influence of the system configuration and vegetation parameters on the value of the vegetation height CRB shows that using an RVoG model with three unknown temporal coherences seems hazardous, since the obtained precision on the vegetation height is usually larger than 6 m for N = 200 pixels. Even with only one unknown temporal coherence, the precision on the vegetation height estimation varies widely as a function of h v and the other vegetation characteristics. For the considered simulations, the precision varies from 0.5 to 10 m for a sample size of 200 pixels. The observed sensitivity of the vegetation height CRB to A, E and h v shows that the description of the problem is complex in dual-baseline. The presented comparison between situations where one or two ground heights are unknown, shows that the difference of performance can remain small. Thus, with phase-calibrated data, it makes sense to use the two proposed estimators and compare their results.

!" #" $" %" &" '" (" )" !" !! !" " !" ! !" # * * !" #" $" %" &" '" (" )" !" !! !" " !" ! !" # * * !" #" $" %" &" '" (" )" !" !! !" " !" ! !" # * * !" #" $" %" &" '" (" )" !" !! !" " !" ! !" # * * !" #" $" %" &" '" (" )" !" !! !" " !" ! !" # * * !" #" $" %" &" '" (" )" !" !! !" " !" ! !" # * * h v [m] h v [m] E = 50m -1 E = 200m -1 E = 2000m -1 E = 50m -1 E = 200m -1 E = 2000m -1 CRB 1/2 N (h v ) (a) (b) h v [m] CRB 1/2 N (h v ) h v [m] CRB 1/2 N (h v ) (c) (d) (e) (f ) h v [m] σ v = 0.023m -1 σ v = 0.046m -1 σ v = 0.069m -1 σ v = 0.001m -1 σ v = 0.023m -1 σ v = 0.046m -1 σ v = 0.069m -1 σ v = 0.
For the analyzed examples, the proposed maximum likelihood estimators seem to be efficient for a sample size of 200 pixels when the polarimetric contrast is larger than 0.6, but often leads to outliers when the polarimetric contrast is smaller than 0.3. This problem may occur because of the initial estimations and therefore the optimization process could be improved. The presented simulations also show that CRBs vary drastically with respect to the ground-to-volume power ratio (from 1 to 5 m), the extinction coefficient (from 1 to 10 m) and the temporal coherence. An obvious perspective is to apply the proposed maximum-likelihood (ML) estimators to real data sets to analyze whether the considered simple model with only one temporal coherence coefficient could be well-suited to extract information from real data with dualbaseline configuration. Since the ML estimator provide estimations of all RVoG parameters, it will be possible to analyze the estimated values of the extinction, the ground heights, and the polarimetric coherency matrices relative to the volume and to the ground. Simultaneously, it will be important to analyse the potential precision of these estimations with the CRB. Another promising perspective is to apply the proposed methodology to other models. In particular, one could investigate the potential of a relatively large number of PolInSAR measurements for biomass parameter estimation, when these are available, to analyze whether such multi-baseline configurations can significantly outperform single-baseline and dual-baseline approaches.

APPENDIX

A. Estimation of θ 0

The algorithm which allows one to get an initial estimation θ 0 is summarized on Figure 6. It consists in exploring the two-dimensional (2D) space formed by the parameters ρ and σ v . For each pair of available baseline,

1 / 2 N 2 N

 122 (h v ) remains larger than 6 m in most cases where 3 temporal coherences are estimated (see (a) and (b)). Nevertheless, the value of CRB 1/(h v ) often lies below 2 m when one estimates only one temporal coherence (see (c) and (d)). When the vegetation height h v varies in the interval [10 m, 50 m], similar images will be obtained and the patterns centered at k z 12 = 2πn/h v and k z 23 = 2πm/h v will partly fill the space (k z 12 , k z 23

N

  } (with p ∈ {1, 2, ..., P }) have been generated according to the considered model (see Eqs. (

Fig. 1 : 2 N 2 N 2 N 2 N

 12222 Fig. 1: Evolution of the vegetation height CRB as a function of k z 12 , k z 23 for N = 200 pixels, A = 0.3, E = 800 m -1 , X = 0.2, β = 35 • , σ v = 0.023 m -1 , ρ = 0.8 and h v = 25 m. (a) CRB 1/2 N (h v ) for θ 25 (i.e. 3 estimated temporal coherences and 2 estimated ground heights). (b) CRB 1/2 N (h v ) for θ 24 (i.e. 3 estimated temporal coherences and 1 estimated ground height). (c) CRB 1/2 N (h v ) for θ 23 (i.e. 1 estimated temporal coherence and 2 estimated ground heights). (d) CRB 1/2 N (h v ) for θ 22 (i.e. 1 estimated temporal coherence and 1 estimated ground height).

  Fig.2: Performances of the vegetation height estimators, as a function of h and for different values of A (A = 0.3 and A = 0.9). In (a) and (c), estimations have been obtained when 2 ground heights are estimated (i.e. θ 23 in Eq. (6)), whereas in (b) and (d), estimations have been obtained when only 1 ground height is estimated (i.e. θ 22 in Eq. (6)). In (a) and (b) the success rate indicates the percentage of h v estimations with an error smaller than 6 m. In (c) and (d) Root mean square errors (RMSE) are computed with estimations for which the error is smaller than 6 m. Straight lines are CRB 1/2

3 .

 3 The other parameters are: k z 12 = 0.06 m -1 , k z 23 = 0.25 m -1 , N = 200 pixels, E = 800 m -1 , X = 0.2, β = 35 • , σ v = 0.023 m -1 and ρ = 0.8.

A = 0. 1 A = 0. 6 A = 0. 1 A = 0. 6 AFig. 3 :

 16163 Fig.3: Same as Figure2, but for A = (large crosses and circles) and A = 0.1 (small crosses and circles).

001m - 1 ρ 2 NFig. 5 :

 125 Fig. 5: Evolution of the CRB with E (a and b), σ v (c and d) and with ρ (e and f). On the left (a, c and e), two ground heights are unknown. On the right (c, d and f), one ground height is unknown. Unless otherwise specified, parameters are k z 12 = 0.06 m -1 , k z 23 = 0.25 m -1 , N = 200 pixels, A = 0.3, E = 800 m -1 , X = 0.2, β = 35 • , σ v = 0.023 m -1 and ρ = 0.8.

  θ 23 = {(t vol,k ) k=1,..,9 , (t gro,k ) k=1,..,9 , z 12 , z 23 , h v , σ v , ρ} with ρ ij = ρ for i = j, θ 24 = {(t vol,k ) k=1,..,9 , (t gro,k ) k=1,..,9 , z 12 , h v , σ v , ρ 12 , ρ 23 , ρ 13 } with z 12 = z 23 , θ 25 = {(t vol,k ) k=1,..,9 , (t gro,k ) k=1,..,9 , z 12 , z 23 , h v , σ

9 , (t gro,k ) k=1,..,9 , z 12 , h v , σ v , ρ} with z 12 = z 23 and ρ ij = ρ for i = j, v , ρ 12 , ρ 23 , ρ 13 },

ACKNOWLEDGMENT

The authors would like to thank Philippe Réfrégier from Institut Fresnel for fruitful discussions and contributions. The authors would like to also thank the CNES (Centre National d'Études Spatiales) for funding the post-doctoral stay of Hélène Sportouche at Institut Fresnel in Marseille, France from 2012 to 2014.

one assumed that ρ and σ v are known and one uses singlebaseline approaches [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF] to estimate potential values of θ.

Once the whole range of ρ and σ v has been covered, the best estimation according to the dual-baseline likelihood criterion is retained. In the considered implementation of this algorithm, the parameter ρ is assumed in the range [0.1, 1] and the parameter σ v is assumed in the range [10 -5 m -1 , 10 -1 m -1 ]. Because implementing a loop with 1000 values for each parameter (ρ and σ v ) is time consuming, we choose to implement 3 successive loops with only 11 values for each parameter to accelerate the process. The detail of these different steps follows.

loop on ρ and σ v

(ρ and σ v assumed to be known)

Fig. 6: Flow chart of the initialization algorithm, which consists in finding the best single-baseline solution with a double loop on the unknown temporal coherence ρ and on the extinction coefficient σ v . The criterion J is the dualbaseline likelihood (see Eq. ( 9)) Firstly, for each baseline ij (with (i, j) ∈ {(1, 2), (2, 3), (1, 3)}), introducing T ij {(i,j)∈{1,2,3} 2 } as the empirical covariance potential ground heights z SBij are estimated using the single-baseline (SB) Cloude estimator [START_REF] Cloude | Three-stage inversion process for polarimetric SAR interferometry[END_REF]. The two intersections in the complex plane between a least square line fit of the three eigenvalues of the contraction matrix [START_REF] Ferro-Famil | Multi-baseline PolIn-SAR statistical techniques for the characterization of distributed media[END_REF] and the unit circle provide two candidates φ SBij for the ground phase of the baseline ij. Clearly, this first step provides several possible estimations of the ground heights. In particular, for θ 22 , there are six available solutions (2 for each pair of baseline), whereas for θ 23 , there are the 8 possible ground height combinations that satisfy equation (4). To remove this ambiguity, one estimate θ for all potential candidates (as detailed in the next paragraph), and finally keep the vector θ that minimize the dual-baseline likelihood criterion J(θ) (see Eq. ( 9)). This criterion is able to remove the ambiguity because it takes into account all baselines.

Secondly, we use a 2D grid on the extinction σ v and on the temporal coherence coefficient ρ, considering 11 potential values σ v in the range [10 -5 , 10 -1 ][m -1 ] and 11 potential values ρ in the range [0.1, 1.0], and we compute the criterion J(θ) for each considered pair ( σ v , ρ). Indeed, for each pair ( σ v , ρ), the estimation of the intersections between the complex function

and the previous line fit (rotated by k z ij z SBij ) provides either zero, one or two candidate vegetation heights, denoted by h ij v for each baseline ij. When this process leads to two candidates, the ambiguity is removed by keeping the one that minimizes the criterion J(θ). Coherency matrices T ij vol and T ij gro are estimated as detailed in appendix B. After parameters have been estimated for each baseline ij, the estimation that minimizes the criterion J(θ) (see Eq. ( 9)) is retained. Furthermore, once the whole 2D grid on σ v and ρ has been covered, the estimation that minimizes the criterion J(θ) is retained. Next, to improve the accuracy of these estimations, a finer 2D grid, new 11 × 11 pairs ( σ v , ρ) but centered on these previous estimations of (σ v , ρ) is used. This refinement of the 2D grid is applied a third and last time.

B. Estimation of T vol and T gro

This section details the estimation of the coherency matrix parameters t ij vol,k k=1,...,9 and t ij gro,k k=1,...,9 from T ii , T jj and T ij when the parameters z SBij , h ij v , σ v and ρ are already fixed. Similarly to [START_REF] Flynn | Direct estimation of vegetation parameters from Covariance data in polarimetric SAR interferometry[END_REF], the coherency matrix estimation problem can be reformulated from equation ( 2) and from equation ( 5) through the following equations:

where the matrices T v and T g and the ratio γ v are defined by:

The estimated matrices T ij vol and T ij gro can be thus deduced from: £

where the matrices T v and T g are determined using:

where (.) is the real part operator.