1 Application of the DORT method to the detection and characterization of two targets in a shallow water wave-guide J-G Minonzio 1 , D. Clorennec 1 , A. Aubry 1 , T. Folégot 2 , T. Pélican 1 , C. Prada 1 , J. de Rosny 1 and M. Fink 1 . 1 Laboratoire Ondes et Acoustique Université Denis Diderot, UMR CNRS 7587, ESPCI 10 rue Vauquelin, 75231 Paris, cedex 05, France 2 ATLANTIDE, Technopôle Brest Iroise, Brest, France *current address: NURC/SACLANT, La Spezia, Italy e-mail adress: jean-gabriel.minonzio@loa.espci.fr Abstract - The decomposition of the time-reversal operator (DORT in French) is an active array detection technique. It requires the measurement of the array response matrix K(ω) and consists in the analysis of the eigenvalues and the eigenvectors of the time-reversal operator K * K which provides information on the presence and localization of scatterers in the medium. It was shown that the DORT method allows to separate and localize pointlike scatterers in a shallow water wave-guide [J. Acoust. Soc. Am. Mordant et al. (1998) and Folégot et al. (2003)]. Here, we extend the study to the detection and frequency characterization of two spherical targets. Small scale ultrasonic experiments are performed with a 3.9 MHz 24 elements transducer array and two spheres of 2 and 3 mm diameter in a 31 mm deep wave-guide. These scatterers correspond to 15 < ka < 25 leading to a non-isotropic scattered field. We have developed a theoretical model taking into account the wave-guide and the acoustic properties of the spheres and using the partial waves decomposition of the scattered field. We calculate the singular values of the array response matrix. This theoretical approach is in good agreement with the experimental results. This 1/325 th scale ultrasonic experiment corresponds to a shallow water experiment with a 12 kHz Vertical Linear Array (VLA). I. INTRODUCTION The decomposition of the time-reversal operator (DORT method) is an ultrasonic detection method that consists in the analysis of the complete set of pulse echo responses of an active array. The eigenvalues and the eigenvectors of the time reversal operator (or the singular values and singular vectors of the array response matrix) provide information on localization of scatterers in the insonified medium. [1] Priors works have shown that the DORT method allows to localize and separate pointlike targets in a shallow water waveguide. [2-5] In this work, we extend the study to the detection and the frequency characterization of two sphericals targets. Small scale ultrasonic experiments are performed using a 3.9 MHz 24 elements transducer array. The wave-guide is 31 mm deep and the bottom is a Plexiglas plate. Two spheres of 2 mm and 3 mm diameters, are immersed at different depth in the wave-guide, 900 mm away from the transducer array. This 1/325 th scale ultrasonic experiment corresponds to an experiment in shallow water with a 12 KHz Vertical Linear Array (VLA) used to localize and characterize two targets of 0.7 m and 1 m diameter at 300 m distance in a 10 m deep channel. In the simple cases, the first and the second singular values of the array response matrix correspond respectively to the strong and to the weak targets at each frequencies. Hence the numerical back-propagation of the singular vectors always focuses on the same target for different frequencies. However, when the apparent reflectivities of the two scatterers are close, crossings occur between the frequency dependent singular values due to each target . In such cases, the back-propagation of the first singular vector focuses on one target or on the other one, depending on frequency. Obviously, the opposite phenomena occurs for the second singular vector. In a theoretical part (section II), the principle of the DORT method is recalled. Then we explained how the anisotropy of the scattered field is taken into account for the calculations of the theoretical singular values and the singular vectors of the array response matrix. Last, the modelisation of the wave-guide is presented. The experimental results are shown in section III. Results obtained with each sphere separately as well as both spheres are presented. II. Principle of the DORT Method First of all, an array of N transmit-receive transducers insonifying a scattering medium is considered as a linear, time-invariant system of N inputs and N outputs. It is characterized at each frequency ω by the array response matrix K(ω). That matrix links the reception vector R(ω) to the emission vector E(ω), with the expression R(ω) = K(ω)E(ω). The length of those vectors is N the number of transducers. So the size of K(ω) is N×N. At a single frequency ω, the K * K matrix is called the time-reversal operator, which is diagonalizable. Its eigenvectors can be interpreted as invariants of the time-reversal process. The eigenvectors of K * K and KK * are the singular vectors of the array response matrix K. [1] The singular values of K are the square roots of K * K (or KK * ). The principle of the DORT method consists in the study of the singular states (values and vectors) of the time-reversal operator, obtained by the singular values decomposition (SVD) of the array response matrix K.