1048 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 4, APRIL 2008

A Coherent Model of Forest Propagation—Application to Detection and Localization of Targets

Using the DORT MethodYoumni Ziadé, Héléne Roussel, Marc Lesturgie, and Walid Tabbara, Member, IEEE

Abstract—We study the propagation of electromagnetic fields inforest media in a frequency band between 100 and 300 MHz. Wedevelop an exact coherent model in order to compute the electro-magnetic field of a received wave after multiple scattering from theelements of the medium, in and outside the forest. We propose sim-plifications for the model of the forest and approximations for theelectromagnetic method to reduce the computing time. We use andsimulate a DORT based method to detect and locate a hidden targetin the medium. We develop this method to adapt it to a real config-uration and we apply it to measurements in anechoic chamber.

Index Terms—Decomposition of time-reversal operator(DORT), detection, experimental results, forest model, mul-tiple scattering.

I. INTRODUCTION

PROPAGATION of electromagnetic waves in forest mediawas the subject of several theoretical and experimental

studies dedicated to the modelling of such medium. When wedeveloped a model to characterize the propagation throughthe forest, the main encountered problem is to include in themodel all the effects of the different elements of this complexmedium. These effects depend largely on the frequency of theincident wave. Therefore, at low frequencies ,the dimensions of the elements of the canopy (small branchesand leaves) are small compared to the wavelength, and theireffects on the propagation are negligible. The forest mediumis modeled by dielectric homogeneous layers representing theground, the vegetation layer (trunks, branches ), the air andthe ionosphere [1], [2]. When we consider higher frequencies

, the elements of the medium are modeledby cylinders (trunks, branches) and ellipsoids (leaves) and thescattering from each element is computed using for exampleradiative transfer approach [3].

Manuscript received February 14, 2006; revised January 16, 2007.Y. Ziadé is with France Telecom, 75505 Paris Cedex, France (e-mail: youmni.

ziade@orange-ftgroup.com).H. Roussel is with the University of Paris VI, 75005 Paris Cedex, France

and also with the Département de Recherche en Électromagnétisme of Supélec,CNRS – Supélec - UPS, F-91192 Gif-sur-Yvette Cedex, France (e-mail: helene.roussel@lss.supelec.fr).

M. Lesturgie is with the Département de Recherche en Électromagnétisme ofSupélec, CNRS – Supélec - UPS, F-91192 Gif-sur-Yvette Cedex, France andalso with the SONDRA Laboratory, F-91190 Gif-sur-Yvette Cedex, France.

W. Tabbara is with the University of Paris VI, 75005 Paris Cedex, France, theDépartement de Recherche en Électromagnétisme of Supélec, CNRS – Supélec- UPS, F-91192 Gif-sur-Yvette Cedex, France, and also with SONDRA Labo-ratory, F-91190 Gif-sur-Yvette Cedex, France.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2008.919150

In this paper, we study the propagation of electromagneticwaves in forest media in the frequency range of 100–300 MHz,where both the transmitting and receiving antennas are close tothe ground. We have developed an exact coherent model thatcomputes the total field received after propagation in the forestat any position, inside or outside the medium. Our model isbased on a domain integral representation of the electric field,allowing us to consider any configurations of the scatterers andfrequency. However, the limitation in frequency is related to theavailable memory space. In practice, the use of a domain integralrepresentation is limited to the analysis of scatterers, the dimen-sions of which are a few wavelengths. In order to model largescenes, we propose two approximated models and compare theirsolutions to the exact one. The electric field calculated with theexact model has been validated and compared with measure-ments in an anechoic chamber [4].

Time-reversal (TR) technique and particularly, TR imagingwith multiple signal classification (MUSIC) method for the lo-calization of targets has proved its efficiency in many applica-tions: medicine, communications, radar and imaging in general.We have applied this technique to our model, first to locate thetrees in the medium, and secondly to detect and locate a targethidden in the medium.

The propagation model we have established has the capa-bility of considering the interactions between the elements (treesor target) of the medium, i.e., the total field is a coherent sumof the waves diffracted by every element in the medium. Wehave apply the decomposition of time-reversal operator (DORT)method to localize and detect target in the medium. We proposealso modifications to simplify the configuration needed to formthe time-reversal operator.

In Section II, we present the full wave 3D method we have de-veloped to model the propagation of electromagnetic waves inthe forest medium and the approximations we have used to sim-plify the model and reduce the computing time. In Section III,we explain briefly the method based on the decomposition of thetime-reversal operator and we apply this method to the forest inorder to determine the positions of the trees. In Section IV, westudy the detection and localization of targets in the medium byapplying the classical DORT method and modified versions inorder to increase its efficiency.

II. THE THEORETICAL MODEL

A. The Forest Model

We have developed a full wave exact method to model thepropagation in forests in the frequency band 100 to 300 MHz.

0018-926X/$25.00 © 2008 IEEE

ZIADÉ et al.: A COHERENT MODEL OF FOREST PROPAGATION 1049

Fig. 1. Forest medium and the forest representation.

We have considered for the representation of the medium all theelements that influence the propagation of the waves. For theconsidered frequency band, the major influence on the propaga-tion is due to the ground, the trunks and the primary branches.The roughness of the ground is small compared to the incidentwavelength, so the reflection from the ground is, in its majorpart, in the specular direction [5]. Thus, we have represented theforest medium by two semi infinite spaces, the ground and theair, separated by a plane interface. Trunks and primary branchesare modelled by vertical and tilted parallelepipeds (Fig. 1).

B. Full Wave Electromagnetic Method

We have developed a full wave 3D model to compute the elec-tromagnetic field everywhere in the forest medium. This modelis based on an exact method that takes into account all the inter-actions between the elements of the medium. The transmittingantenna is placed in the air and close to the ground at the entryof the forest and we study the received wave after its propaga-tion through the medium.

We can decompose the method in three steps. In a first step,we compute the reference field at the positions of the trees: thisis the field radiated by the transmitting antenna in the refer-ence medium (the layered medium in absence of the trees). In asecond step, we compute the total field inside the trees, takinginto account all interactions. And finally, we compute the re-ceived field, which is composed of the field radiated by the trans-mitting antenna and the field diffracted by the trees.

The reference field depends on the transmitting antenna. It in-cludes the field of the direct wave and that of the wave reflectedfrom the ground.

The total field in the trees is computed using the integral rep-resentation of the field given by

(1)Where is the total field at the point r in the air layer,

is the reference field, is the permittivity contrastbetween the scatterers (the trees) and the air and is theGreen’s function of the layered medium [6], [7]. The integralis over the volume containing the trees . The second termon the right hand side part of (1) represents the diffracted field.To solve (1) and compute the total field in the trees, we use amethod of moments [8]. We discretize the trees in cubic cellsof size where is the wavelength in the trees. Weobtain a system of 3 linear equations given by

(2)

where is the number of cells necessary to discretize all thetrees, defines the position of the th cell of atree. The computing time needed to solve this system becomesvery large when we consider many trees. In order to reduce thecomputational cost, we introduce and validate simplificationsdescribed in the next paragraph.

The received field is obtained by using (2) when

(3)where is the position of the receiver.

C. Simplifications to Reduce the Computing Time

The model described can consider all multiple interactionsbetween the incident wave and the elements of the medium. In(2), the total field in a cell is linked to the fields in all the cellsof the medium. The matrix representation of (2) is

(4)

where and are 3 -vectors,

and .

is a 3 3 matrix and are 3 3

matrices ( , ). To solve (4), an inversion of matrixis needed, leading to expensive calculations and high memoryallocation, hence some approximations are used.

In general, the matrix is diagonally dominant, i.e., thetotal field in a cell is mostly influenced by the incident fieldin the cell itself. The influence of another cell depends on itsdistance to the cell and on the frequency of the incidentwave. We propose two simplifications in order to reduce thecomputing time. A first method consists in neglecting thecoupling between the trees when their separation is sufficientlylarge compared to the wavelength. In this case, we computethe total field in each tree independently, only the couplingbetween the cells of a same tree is considered. The sum in(2) is now over all the cells of a tree and becomes a 3

3 matrix where n is the number of cells of the consid-ered tree. This method (called NC) reduces significantly thecomputing time and allows us to consider a large numberof trees. The second method is based on the extended Born(Ex-Born) approximation [9]. An approximate form of matrix

is obtained by replacing in each row the diagonal element byits sum with the non diagonal elements. We obtain a new ma-trix: . iseasily inversed and the total field is given by the following.

The basic principle of the Ex-Born approximation is that ma-trix is diagonally dominant for low frequencies. When thefrequency increases, the diagonal dominance becomes weakerand the approximation becomes less efficient.

D. Results of Simulation

We consider a medium composed of four trunks of diameter0.5 m and height 5 m (Fig. 2). The coordinates of the centres of

1050 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 4, APRIL 2008

Fig. 2. A top sight view of the medium.

the trunks are (2 m,1 m), ( 2 m, 2 m), ( 3 m, 2 m), (1 m, 2m). The relative permittivities of the trunks and the ground are

and , respectively.The transmitter antenna is an elementary vertical electric dipoleplaced at 100 m from the origin at a height of 0.75 m. The re-ceivers are placed over an arc of radius 100 m. We compare theelectric field diffracted from the medium, computed using theexact method and the both approximations proposed in C, at thereceiver’s positions between and , at threeincident frequencies: 100, 200, and 300 MHz.

We can see on Fig. 3 that the Ex-Born approximation is moreefficient at low frequencies. When the frequency increases, theratio of the distance between trees to the wavelength increasesand the effects of the coupling decreases. The NC approxima-tion becomes more efficient.

E. Effect of Branches

To study the effect of the branches, we consider 4 treescentered on (2 m, 3 m), (5 m, 2 m), (3 m, 2 m) and (6m, 3 m). Each tree is formed by a trunk of diameter 0.5m and height 4 m, and three branches of length 0.5 m anddiameter 0.2 m. The transmitting antenna is placed on (0, 0,2 m). We show on Fig. 4 the electric field which is receivedover a circle of radius 7 m and centered on (4 m, 0, 2 m).We can see that the effect of branches on the diffracted fieldis negligible for the considered frequency band. In the studyof detection and localization of targets in the medium, wewill neglect this effect and the trees will be modelled byvertical parallelepipeds representing the trunks. Indeed, in theconsidered frequency band, the trunks are the major elementsof the trees that influence the propagation [4], [5].

III. LOCALIZATION OF TREES IN THE MEDIUM

In this section, we study the possibility of retrieving the po-sitions of trunks in the forest using the DORT method. De-tails about this method are given in [10]–[14]. Here we explain

Fig. 3. Module in V/m (left) and phase in rd (right) of the vertical component ofthe electric field at the receivers computed using the exact method (solid line),the Ex-Born approximation (dotted line) and the NC approximation (dashedline), for 3 incident frequencies. (a) fr = 100 MHz; (b) fr = 200MHz; (c)fr = 300 MHz.

briefly the basics of the method and we present some results ofsimulation.

ZIADÉ et al.: A COHERENT MODEL OF FOREST PROPAGATION 1051

Fig. 4. Comparison of module in V/m (left) and phase in rd (right) of thevertical component of the electric field obtained by considering the effect ofbranches (solid line) and by neglecting these effects (circle), for 2 incident fre-quencies. (a) fr = 100 MHz; (b) fr = 300MHz.

A. The Time-Reversal Operator

In our case, we consider an array of transmitters/receiversantennas and the medium composed of a limited number oftrunks. Each antenna transmits a wave at a fixed frequencyand the wave scattered from the medium is measured and reg-istered at each antenna of the array. We obtain the multistaticdata matrix , where the element isthe field of the wave diffracted from the medium received at theth antenna when the th antenna transmits. The reference field

(due to the propagation without the trunks) is not included in theexpression of the elements of matrix .

The time-reversal operator TRO is defined by .Let and be the eigenvectors and theeigenvalues of matrix . It was proved [10] that for greaterthan the number of targets in the medium, and for the case ofwell-resolved targets, the number of non zero eigenvalues is

equal to the number of targets in the medium and moreover, eacheigenvector of non zero eigenvalue provides the phase law to beapplied to the antennas in order to focus on one of the targets.Let us note that the objects we want to locate in our applica-tion (trees or targets) are composed of many scattering points(each cell of the object is an elementary source that diffractsto the array), and the cells cannot be well-resolved because ofthe effect of the multiple interactions taken into account in ourmodel. These multiple interactions between the cells are con-sidered even when we eliminate coupling between trees; in thiscase they are reduced to the interactions between cells of thesame tree. Therefore, we are in the case of non-well resolved tar-gets and in such case, a MUltiple SIgnal Classification (MUSIC)type algorithm can be employed and gives good estimations ofthe targets location.

The matrix defines a vector space, , of the electromag-netic fields of the waves received at the array after propagationin the forest medium. In the case of well resolved targets, thisvector space can be decomposed into two orthogonal subspaces:the first one is the signal subspace generated by ,and the second one is the noise subspace generated by

, (note that practically, the lowest eigenvalue corre-sponds to the level of the uncorrelated noise)

and

where is the number of targets. Letbe the vector of the fields radiated from the

antennas at a point in the reference medium (without anyscatterer). If corresponds to a target position will beperpendicular to the subspace , and we can write

In other words, the subspace is the subspace generated bythe vectors where is the th target’sposition.

We form the pseudo-spectrum

(5)

where is a test target location. We can conclude thatwill become large at all target locations and will be small other-wise.

B. Results of Simulation

We applied the DORT method to study the localization oftrees in a forest medium and then test the efficiency of themethod proposed in A. We consider the medium described inSection I (Fig. 2). One hundred transmitter/receiver elementaryvertical dipoles are placed over a circle of radius 100 m centredon the origin ( , ) of the medium at a height of 0.75

1052 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 4, APRIL 2008

Fig. 5. (a) Eigenvalues obtained using the exact method (solid line), theEx-Born approximation (dotted line) and the NC approximation (dashed line).(b) The 50 last eigenvalues.

m. The frequency of the emitter is equal 300 MHz (single tone,no bandwidth).

We studied the effect of the approximations we have proposed(NC and Ex-Born approximations) on the localization of the ob-jects in the medium. For that purpose, we form three TRO usingthe full wave 3D method and the two approximations respec-tively (NC and Ex-Born) and we apply the DORT method toeach operator. We present in Fig. 5(a) the eigenvalues obtainedfor the three operators. We can notice that there are about twentynonzero eigenvalues. The other values are small compared to thefirst twenty values and they decrease rapidly, but they are notexactly equal to zero. This is due to the multiple interactionsbetween the cells. In fact, a wave at a receiver is the sum of twocontributions: the first one is that of the waves directly scatteredby the cells toward the receiver and the other one is that of thewaves reaching the receiver after multiple diffractions. The first

contribution is the one, and the first eigenvalues correspond tothis part. The second contribution can be considered as an ad-dition noise that perturbs the propagation and the lowest eigen-values correspond to this part. Let us compare the eigenvaluesobtained by the three methods. An interesting result is observedin Fig. 5(b) where we plotted the fifty last eigenvalues. We cansee that the exact method gives greater eigenvalues than the twoother approximations. This can be explained by the fact thatwhen we apply the exact method, the effect of multiple interac-tions is important compared to the cases where we apply one ofthe approximations, and the noise (due to multiple interactions)is higher in this case. Comparing the two approximations leadsus to a similar conclusion: in the case of NC approximation, themultiple interactions between the cells of a tree are taken into ac-count and the noise is present, but less important than in the caseof the exact method; in the case of the Ex-Born approximation,these multiple interactions are not considered and so the eigen-values in this approximation are smaller than in the two othercases.

At this point, we have explained why we do not obtain exactzero eigenvalues, but we did not give a response to the impor-tant question: starting from which value we can consider thatthe eigenvalues are null? We need for this value to compute thepseudo spectrum . We can, in a first step, rewrite the signaland noise spaces as follows [10]:

and

where is a small threshold parameter. Under these conditions,we can redefine the pseudo spectrum as follows:

(6)

We compute this redefined pseudo spectrum by consideringeach of the three methods. The threshold parameter is fixed sothat . The test points are located on a horizontalplane at an altitude of 0.75 m. In Fig. 6 we show the pseudospectrum in dB obtained in the areaand . The three methods give very closespectrums. For that, we choose to represent only the spectrumobtained by applying the exact method. We can see on Fig. 6that the trees are well-localized, but the level of the noise inthe image is not negligible, and it exceeds in some positionsthe level of the spectrum at the trees positions. To amelioratethe quality of the image, we propose a new expression of thepseudo-spectrum defined by

(7)

ZIADÉ et al.: A COHERENT MODEL OF FOREST PROPAGATION 1053

Fig. 6. Pseudo-spectrum P in dB obtained for a frequency fr = 300MHz.

Fig. 7. Pseudo-spectrum Q in dB obtained for a frequency fr = 300MHz.

The modified pseudo-spectrum is a multiplication of thepseudo-spectrums formed by increasing the threshold param-eter. The expression of is based on the idea that the eigen-values are classified by descending order, so that the last eigen-value corresponds to the smallest one, i.e., the last eigenvectorhas the smallest projection on the subspace . We show onFig. 7 the pseudo-spectrum applied to the TRO obtained forthe exact model and we can notice that the level of the noiseis reduced, and the identification of the trees in the image ismore obvious. Let us note also that the dynamic of the spec-trum is much larger than that of the spectrum

, which facilitates the identification of the objectsin the medium.

The DORT method associated with the MUSIC algorithm,and specially the modified pseudo- spectrum , has proved itsefficiency to localize clearly the objects in the medium indepen-dently of the method used to obtain the TRO. In the next section,we will study the detection and localization of a metallic target

Fig. 8. Top sight view of the medium with the metallic target.

placed among the trunks. For that purpose, we will use the ap-proximate models to compute the TRO to avoid using the exactmethod which needs a large computing time especially if weconsider a large number of objects in the medium.

IV. DETECTION AND LOCALIZATION OF TARGETS

We consider a target situated in a time-invariant clutteredmedium and an array of transmitter/ receiver antennas placedover an arc. Two measurements are performed to detect a targetand to locate it. First, we measure the cluttered background envi-ronment in the absence of the target. We obtain a first multistaticdata matrix . Then, we measure the medium in the presenceof the target. We obtain a second multistatic data matrix .The nonzero nature of the difference between and in-dicates a change in the medium under test [15]. This differencerepresents mainly the response of the target and the effects ofmultiple scattering between the targets and the trees. We applythe DORT method and MUSIC algorithm to this difference tolocate the target in the medium.

A. The Conventional DORT Method

We consider the cluttered medium composed of four trees de-fined as in Section II. The target is a metallic cube

placed on the ground at ( , )(Fig. 8). Twenty elementary vertical dipoles are regularly placedover a 40 arc of radius 20 m. We apply the Ex-Born approx-imation to compute the matrices and . The test pointsare located, as previously, on a horizontal plane at an altitude of0.75 m. We show on Fig. 9 and Fig. 10 the pseudo-spectrums

and related to at the frequency .The conventional pseudo-spectrum gives an erroneous local-ization of the target and the pseudo-spectrum eliminates falsedetection and gives the correct location of the target. We mustnote that the difference between levels of at the target posi-tion and other high level observed on Fig. 10 [in particular atthe point ( , )] is greater than 30 dB. The

1054 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 4, APRIL 2008

Fig. 9. Pseudo-spectrum P in dB computed for the case of the target in themedium—fr = 100 MHz.

Fig. 10. Pseudo-spectrum Q in dB computed for the case of the target in themedium—fr = 100 MHz.

localization for other frequencies was tested and gave also goodresults for .

B. Separation of the Transmitter and the Receiver Arrays

One limitation of the DORT method lies in requirements ofconsidering multiple positions of the antennas to form the ma-trix , which is not always easy to realize in practice, especiallywhen transmitter and receiver antennas coincide. In this case,the coupling between antennas influences the received field, andby separating the transmitter and the receiver arrays we canavoid this problem. In [16], the DORT method and MUSIC al-gorithm are generalized to the case where the transmitter andreceiver arrays do not coincide. Let and be the num-bers of transmitters and receivers respectively. The multistaticdata matrix is a matrix. Let and

Fig. 11. Transmitter and receiver arrays are separated.

. and are the eigenvec-tors and eigenvalues of the matrices and respectively. Thepseudo-spectrum is now defined as

(8)

where is the vector of thefields radiated by the transmitting array at the test pointand is the vector of the fieldsradiated by the receiving array at the test point .will have peaks at the scatterers location. The expression (8)combines the transmitters and receivers with and appliedto the transmitters and receivers respectively.

We apply this method to detect the target for the same config-uration as the one defined previously. Twenty transmitters andtwenty receivers are placed over two arcs of aperture 40 andof radius 30 m (Fig. 11). We use the approximation NC to com-pute matrices and . On Fig. 12, we show the spectrum

for the frequency of . The target is detectedand well localized in the medium.

C. Transmission of a Wide Frequency Band Signal

Separating the transmitters and the receivers can simplify andmake easier the measurements in some configurations, but insome other cases, it is not simple to consider many positionsof the antennas. We propose to reduce the number of transmit-ting antennas, the reception remains over antennas. To fillthe lack in the transmission, we transmit multiple frequencies

, where is the number of transmitted frequen-cies and we register the received field at the receiver array ateach frequency. We obtain a new matrix ( elements).The TRO is a square matrix. The TRO

is a square matrix. and

ZIADÉ et al.: A COHERENT MODEL OF FOREST PROPAGATION 1055

Fig. 12. Modified pseudo-spectrum H in dB—fr = 300 MHz.

are the eigenvectors and eigenvalues of the ma-trices T and L respectively. We propose a new definition of thepseudo-spectrum

(9a)

(9b)

(9c)

where is a vector of the fields generatedby the receiving antennas at a test point at the frequency

. is a vector. Each element of thisvector can be written as the product of to elements:

. is the field of the wave generated by thetransmitting antenna at the test point at the frequency .

is the field of the wave generated by the receivingantenna at the test point at the frequency .

We consider the same medium and target as previously. Thetransmitting antenna is placed at 30 m far from the centre of themedium. Sixteen frequencies regularly spaced between 150 and300 MHz are transmitted. The receiving array is placed over a40 arc of a radius 30 m. The new pseudo-spectrum is showedon Fig. 13. We can see the efficiency of this new spectrum inestimating the position of the target.

D. Application to a Forest Scaled Model With Hidden Target

Finally, we had the opportunity to apply the MUSIC algo-rithm to fields measured in an anechoic chamber. The measure-ments were done at the Office Nationale d’Etudes et RecherchesAerospatiales (ONERA), in a bistatic configuration, on a scaledmodel of periodical forest. The scale factor is 1/50. The medium

Fig. 13. New pseudo-spectrum R in dB.

Fig. 14. Photo of the measurement.

TABLE IPARAMETERS OF THE MEDIUM

is composed of 8 8 vertical parallelepipeds distributed regu-larly on a metal plate (Fig. 14). We show in Table I the param-eters of the medium in the real case and in the scaled model.

The transmitting antenna is a hornantenna ofdimensionsand the transmitted wave is vertically or horizontally

polarized.The receivingantenna is similar to the transmittingoneand can move on an arc of radius 5.2 m centred on the centre of

1056 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 4, APRIL 2008

Fig. 15. Top sight view of the medium with the metallic target.

the considered medium. The angle of reception varies from 150to 190 with a step of 2 . The electric field is measured at thepositions of the receiver and for the 38 transmitted frequencies(between 7.5 and 15 GHz). We obtain the multistatic matrix .Comparisonbetweentheresultsobtainedfromthemodelwehavedeveloped and from the measurements is given in [4].

A metallic parallelepipedic object, of length 26 cm, width 4 cmand height 4 cm, is placed in the medium as shown in Fig. 15. Asecond measurement of the medium with the target gives the mul-tistatic data . We deduce the matrix , which isthe response of the target and the multiple interactions betweenthe target and the medium. Both VV and HH polarizations aremeasured and used to compute the TRO and the pseudo-spectrum

defined in (5). In Fig. 16(a) and (b), we show for VV andHH polarizations, respectively. We can see that HH polarizationgivesagoodestimationof thepositionof the target in themedium.However, VV polarization gives a bad result of localization of thetarget. This can be explained by the following: in the case of VVpolarization, the effect of the vertical trunks on the propagationis important compared to the case of HH polarization, and sim-ilar are the effects of the multiple interaction of the wave betweenthe target and the trunks. On the other hand, the response of thetarget is higher in HH polarization than in VV polarization, be-cause of the horizontal geometry of the target. These two pointsindicate that in the case of HH polarization, the received field is,in its major part, diffracted directly from the object, but in the caseof the VV polarization, the received field is mainly influenced bymultiple interactions and the received wave has underwent mul-tiple scatteringbefore reaching the receiver. Otherwise, the radialresolution is good, the transverse one is weak because it dependson , the variation of the angle of reception, and in the givenexample is small . To improve the transverseresolution and the quality of the image we should measure thediffracted field on a larger range.

V. CONCLUSION

In this paper, we have presented a coherent model using theintegral representation of the electric field to study the prop-

Fig. 16. Pseudo-spectrum R in dB obtained for (a) horizontal and (b) verticalpolarization.

agation of electromagnetic waves in forest media. The modeltakes into account multiple interactions between the wave andthe trees in the medium. We have proposed two approximatemodels in order to reduce the computing time. The first one (NC)eliminates the coupling between trees, the computing time de-pends on the number of cells used to discretize each tree and noton the total number of cells, which makes it possible to considera large medium. The efficiency of this approximation dependson the frequency and the distance between the trees. The secondmodel is based on the Ex-Born approximation which is efficientat low frequencies. Higher orders based on the Ex-Born approx-imation are given in [7].

We have applied the DORT method associated to MUSIC al-gorithm to detect and locate targets in the medium. First, wehave verified the conventional DORT method to locate the treesand to detect and locate a target in the medium. Then, we haveproposed to reduce the number of transmitters and to transmit awide band signal. We have proved the efficiency of this config-

ZIADÉ et al.: A COHERENT MODEL OF FOREST PROPAGATION 1057

uration in locating targets by showing results of simulation andmeasurements in anechoic chamber.

ACKNOWLEDGMENT

The authors acknowledge A. Cheraly and F. Tardivel fromONERA for their contribution to the study, in particular for theperformance of the anechoic chamber measurements.

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Youmni Ziadé was born in Tripoli, Lebanon, in1980. He received the Diploma in computer andcommunication engineering from the LebaneseUniversity, Beirut, Lebanon, in 2002, the M.S.(DEA) degree from the University of Rennes 1,Rennes, France, in 2003 and the Ph.D. degree fromthe University of Paris VI, Paris, France, in 2006.

From 2003 to 2006, he worked in the “Laboratoiredes Signaux et Systèmes” at SUPELEC France on thepropagation of electromagnetic waves in forest mediaand on time reversal applied for detection and local-

ization of targets in inhomogeneous media. Since march 2007, he has worked,

as a postdoctoral fellow, at France Telecom RD, Paris, France. His research in-terests include the application of time reversal technique in communicationssystems.

Héléne Roussel was born in Courbevoie, France, in1967. She received the Ph.D. degree from the Univer-sity of Paris VI, Paris, France, in 1993.

Since 1995, she has been an Associate Professor atthe University of Paris VI, her research is conductedin the Département de Recherche en Électromag-nétisme of Supélec. Her research interest is in thearea of waves in inhomogeneous media for variousapplications, absorbing structures, photonic crystalsand remote sensing.

Marc Lesturgie graduated from the Ecole Na-tionale Supérieure de l’Aéronautique et de l’Espace(ENSAE), Toulouse, France, in 1985 and receivedthe Master degree (DEA) in electronic and mi-crowave from the University of Toulouse, in 1986and the Research Directorship Habilitation from theUniversity of Paris VI, Paris, France, in 2005.

In 1987, he joined the French research centerOffice National d’Etudes et de Recherches Aérospa-tiales (ONERA) as a Research Engineer and workedin a wide range of low frequency and new concept

bistatic and multistatic radars. From 1996 to 2000, he was the Head of the“New radar concepts” team. Currently, he is the Head of the Département deRecherche en Électromagnétisme of Supélec and the Director of the SONDRALaboratory, Gif-sur-Yvette, France, a joint laboratory established betweenSupelec, ONERA, and the National University of Singapore and the DefenceAgency of Singapore. He is the author and coauthor of over 30 journal andconference papers, several patents, and over 40 technical reports

Mr. Lesturgie is the Chairman of the SEE/Committee 23 (radio-location andnavigation), has organized several international conferences, and acted as theTechnical Chairman of the International Conference on Radar Systems in 2004.

Walid Tabbara (M’83) was born in Beirut, Lebanon,in 1946, he received the Doctorat ès Sciences (Ph.D.)degree from the University of Paris VI, Paris, France,in 1976.

He has been a Professor at the University of ParisVI since 1981. His research is performed at theDépartement de Recherche en Électromagnétisme ofSupélec. From 1987 to 1996, he acted as a DeputyDirector of the Laboratoire des Signaux et Systèmes(Supélec-CNRS). Since 2004, he also acts as Tech-nical Manager of the French-Singaporian laboratory

SONDRA, Gif-sur-Yvette, France. After working for a long period in the fieldof inverse scattering in electromagnetics and acoustics, his present work fo-cuses on EMC, remote sensing and propagation in complex environments. Hispresent and past research has led to the publication of 50 papers in internationaljournals and to 180 presentations at international and national conferences.

Dr. Tabbara is a member of URSI, the Electromagnetics Academy and ofthe editorial boards of the JEWA and of the Journal of Computer Applicationsin Engineering Education. He acted as the French delegate to Commission B ofURSI from 1984 to 1990. He organized and/or chaired sessions at national andinternational conferences.