Singularity Characterization by Monoscale Analysis: Application to Seismic Imaging

Seismological images represent maps of the Earth's structure. Apparent bandwidth limitation of seismic data prevents successful estimation by the multiscale wavelet transform of Lipschitz/Hölder regularity of nonoscillating singularities. To overcome this fundamental problem, a new method is proposed which provides local estimates from information, essentially residing at only one single scale. Within this method, the exponents are no longer calculated from the decay or growth rate of the wavelet coefficients. Instead, the estimates are obtained by transforming the data with respect to a family of generalized “wavelets” of fractional order. This generalized family is defined in terms of causal and anticausal, fractional integro-differentiations of a fixed-scale, Gaussian smoothing function. Supplementing this transform with criteria that predict the onset or disappearance of modulus maxima as a function of the wavelet order, we provide exponent estimates at the scale of the smoothing function. The estimated exponents are equivalent to Hölder exponents when the scale of the smoothing function approaches zero.