1. An apparatus that extracts slowness dispersion characteristics of one or more sonic wave forms in broadband acoustic waves received by an array of two or more sensors positioned with respect to a reservoir, the apparatus comprising: a circuit that digitizes the sonic wave forms to form discrete time wave forms and converts the discrete time wave forms into frequency domain wave forms data; a circuit that divides a processing band of the frequency domain wave forms data into frequency sub-bands data; one or more processors that, for each sub-band: approximate a family of candidate dispersion curves for multiple modes; parameterize each of the family of approximated candidate dispersion curves by phase and group slowness; form a frequency dependent over-complete dictionary of basis elements, each corresponding to a pair of phase and group slownesses, and spanning a range of values thereof; form multiple measurement vectors from the frequency domain wave forms data; implement a sparse Bayesian learning (SBL) algorithm on the multiple measurement vectors with a block sparse signal model by: estimation of a mode spectrum amplitude with a Bayesian approach; estimation of a mode spectrum variance, noise variance and other model parameters with a maximum likelihood estimation; assessment of presence of spectrum correlation patterns of multiple modes in the frequency domain wave forms data based in part on an estimation of covariance therein; prune mode candidates based at least in part on the assessment and by comparison of the estimated mode parameters with a threshold; iteration between the Bayesian approach and the maximum likelihood estimation until convergence according to a criterion; and output the phase and group slowness and the mode spectrum amplitude for a given sub-band; and that generate a final dispersion curve over the processing band; and characterize the reservoir based at least in part on the final dispersion curve.