Spectral Analysis of Randomly ScatteredSignals using the Wavelet Transform

Estimation of power spectra is a central question in signal analysis and its applications such as Geophysics. This is classical for stationary signals for which the power spectral density, i.e. the Fourier transform of the covariance, is deened as a function of the frequency only. In the general case, signals under study are not stationary. This work has been motivated by signals obtained in the context of wave propagating in random media. A one-dimensional model of random media is given by a layered medium whose properties are random variables in each layer independent from one layer to another. It has been shown ,,1], that if a pulse of typical wavelengths is sent in such a medium with small layers of order 2 , the reeected signal is a non-stationary stochastic process which converges in distribution, as goes to 0, to a locally stationary gaussian process. This convergence will be the key property in the estimation of the power spectrum for the reeected signal. We introduce an estimation of the local power spectrum using wavelet transform. In our main theoretical result we identify the coeecients to be computed and we present the corresponding algorithm with numerical experiments. In the last section we show how this wavelet method enables us to get a fairly good estimation of the small parameter when this one is not known a priori.