Fractional Brownian Motions (fBm) are nonstationary and selfsimilar stochastic processes which extend ordinary Brownian motion and are of great importance for modeling processes with long-term dependencies, such as l/f-type processes. Identification of fBm amounts to estimate one single scalar parameter : the fractal dimension, related to the roughness of fBm's samples. According to the structural properties of fBm, different fractal dimension estimators can be considered. We have chosen five of them, which operate either in the frequency domain (identification of a spectral exponent via spectrum analysis), in the time domain (maximum likelihood on one hand, methods based on length measurements of fBm's samples at different observation scales on the other hand) or in a mixed time-scale domain (identification of a self-similarity parameter via the variance of wavelets coefficients). The relevance of these differents estimators is discussed and their performance is compared on simulated and real data.I -FRACTIONAL BROWNIAN MOTIONS I n a number of physical phenomena, strong long-term dependencies are involved and l/fp spectral behaviors are observed over a wide range of frequencies [ 11. A convenient modeling for processes of this kind has been proposed by Mandelbrot and Van Ness [2] and is referred to as fractional Brownian motion (fBm). By definition, fBm is a nonstationary process whose expression reads (S) .P + j ( t -s)H-l/*dB(s)); 0 < H < 1, where B ( t ) = B,/2(t) is ordinary Brownian motion. The increments of fBm are stationary and, moreover, they are statistically self-similar, which means that for any a > 0 (and with the convention BH(0) = 0), [ Btl(at)) and [ aHB&)) undergo the same probabilistic behavior.In the frequency domain, an average power spectrum density (PSD) of fBm can be defined [3] and it tums out that it is proportional to I/fp with p = 2H + 1. In the time domain, the selfsimilarity structure which underlies the definition of fBm allows to associate to it a fractal dimension D = 2 -H , which varies therefore between 1 and 2 [2]. This fractal dimension is related to the roughness of fBm's samples and it appears as a relevant parameter for identification and classification [4], motivating hence the development of efficient procedures aimed at its estimation. 0 * fomcrly with LTS-ICPI Lyon # formerly with EDFIDER'RISDM t also with G D R 134 CNRS "Trarlemenl du Signal el Images" I1 -FRACTAL DIMENSION ESTIMATORS According to the above properties, five different estimators of D have been considered, each based on some specific feature of fBm. A. Spectrum AnalysisThis exploits the fact that, because fBm has a llfp spectral behavior, its PSD is supposed to be a straight line in a log-log plot. The slope -b of this line can therefore be estimated by a least-squares fit on some log-log periodogram estimate ( Fig. 1) and D can be deduced from pas D = (5 -m/2.It is clear that such an estimator necessitates a large amount of data points for achieving a significant variance reduction. Another fundame...

This paper presents a new sonar target classification approach based on the use of time-frequency filters. Their design is carried out from the free field response of a reference target, and more precisely from the analysis of echo formation mechanisms in the time-frequency plane. The study of the relevance and the robustness of this approach in approximately real sonar conditions is conducted from experimental measurements in a tank. A data base is set up that contains a large set of target responses in the free field, near different interfaces and in waveguide situations. First, the efficiency of the method for the recognition of a nickel molybdenum spherical shell, corresponding to a class of man made targets whose size is much smaller than the sonar beam (finite size) is shown (100% of recognition). Second, a classification procedure between different targets of finite size is conducted: more than 85% of good classification is obtained (except for the marble solid target). Finally, in the presence of numerical noise, the method is found to be robust even for a low signal to noise ratio.

Efficient sonar target recognition must be based on a set of relevant parameters extracted from the echoes and directly related to the geometrical and mechanical properties of the target. Previous studies have shown that a wideband approach based either on simple spectral modeling or on time-frequency analysis is particularly suited to the understanding of echo formation mechanisms and to the classification in the free field (recognition rate as high as 96%). In shallow waters, the influence of the boundaries considerably reduces the performance of these methods due to the presence of several spurious echoes superimposed to the target echo components. Classification performance can be improved by using a method based on time-frequency filtering matched to a free-field reference target response. This method will be described and applied to experimental data issued from tank experiments. Several typical configurations have been first investigated: free-field, semi-infinite space for a set of targets (shells and solid targets of different shapes). It gives more than encouraging results, especially for the recognition of a spherical shell (about 100% of recognition). The method is now extended to a shallow-water configuration including seabed interaction.

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